Numerical Simulation and Application of Vortex Field Monitoring near Islands in Straits

Abstract

This study addresses the issue of vortex current fields near the islands and reefs in China’s straits, which pose significant challenges to engineering construction and navigation safety in the surrounding waters. To monitor these vortex fields, the study proposes an innovative method utilizing acoustic signals. The study utilizes the numerical simulation and phase feature extraction of acoustic signals in the vortex current field, based on ray acoustic theory and time-reversal mirror technology. The study successfully monitored the central position of the vortex core and characteristic radius of the vortex current field near Barley Straw Reef using HYCOM data for the first time. Furthermore, the performance of the method was analyzed under different acoustic phase perturbations and signal-to-noise ratios. The numerical simulation results demonstrate that the acoustic method is effective in monitoring near-shore vortex fields, and the time-reversal mirror technique is useful in extracting phase difference information from acoustic signals generated by vortex currents.

1. Introduction

China has a vast sea area with numerous islands, and vortex current fields are often induced near the channel islands by high and low tides [1]. The spatial scale of the vortex field near islands and reefs ranges from 0.1 to 10 km, with a time scale of hours to days [2]. Despite its small temporal and spatial scales, the presence of a vortex current field poses a safety hazard to near-shore construction works and increases the risk of collision for moored vessels. This highlights the importance of a monitoring study of the vortex current field near the islands and reefs in the straits. Such a study can lay the foundation for scientific guidance and decision-making for the construction and development of offshore projects, as well as ensuring navigational safety for ships in harbors.

With the progress of science and technology, various methods have become available for monitoring ocean vortex currents, including satellite remote sensing technique [3,4,5], Argo profiling floats [6,7,8], and ocean acoustic tomography [9,10,11]. However, each of these methods has its intrinsic limitations. For instance, the monitoring based on satellite remote sensing is not real-time; Argo profiling floats require manual round-trip measurements that can be labor-intensive; and ocean acoustic tomography is costly. It has been observed that when acoustic waves travel through a vortex current field, there is a significant change in their amplitude and phase [12]. This phase information can be utilized as a feature to determine the presence of vortex currents, leading to the development of a monitoring method based on the phase eigenvalue of acoustic signals. This method is cost-effective and available for real-time dynamic monitoring in comparison to the aforementioned techniques.

The scale of the vortex current field near an island is generally much larger than the wavelength of the acoustic waves. As a result, the main effect of vortex current on sound propagation is mainly manifested in refraction. Therefore, the ray acoustic model is suitable for simulating and calculating acoustic signals. Over the past few decades, several scholars have extensively studied acoustic propagation in vortex current fields using ray acoustic models. For example, Yu et al. [12] studied the mapping relationship between the phase of the received acoustic signals and the characteristic parameters of the vortex field based on ray acoustic theory. Bedard et al. [13] used ray acoustics to explore the impact of atmospheric hurricanes on infrasound refraction. The results of the study showed that the hurricane field can change the propagation trajectory of the ray, while the impact of the temperature field was minimal. Torres et al. [14] reconstructed the plane wave based on ray acoustic theory to analyze the wavefront surface of scattered waves from the vortex field. Jiang et al. [15] investigated the influence of the vortex on the vertical spatial characteristics of wind-forming noise using the ray equation model. Despite the listed studies, studies on the numerical simulation calculation of acoustic signals in vortex current fields are still rare, particularly in terms of sound pressure amplitude. The lack of accurate sound pressure data restricts the effective simulation of the acoustic signals at the receiving point, which in turn limits further research on the feature extraction of vortex current fields.

Feature extraction is crucial in monitoring the vortex current field near islands and reefs. In particular, the accurate extraction of the acoustic phase is of utmost importance. However, at a low Mach number, the change in acoustic signals in the vortex current field is minimal and often interfered with by the marine environment. This further complicates the extraction of acoustic signal phase features. Previous studies proved that the time-reversal mirror (TRM) technique can be used to amplify the phase changes in acoustic signals in vortex current fields, thus easing the feature extraction. For instance, Roux et al. [16] demonstrated the effectiveness of the double TRM in amplifying the phase change in acoustic signals in vortex current fields through ultrasonic experiments. Manneville et al. [17,18] also utilized TRM to study the phase change in vortex acoustic signals by amplifying the impact of vortex currents on acoustic signals and successfully applying it to the extraction of phase features in vortex acoustic signals. However, these findings were obtained using laboratory settings, and the application in the marine environment has yet to be explored.

Despite the results obtained on the propagation of acoustic waves in vortex current fields, the calculation of acoustic wave amplitude in these fields remains imperfect, let alone the fact that the application of the techniques previously is under laboratory settings instead of marine environments. Therefore, this paper aims to refine the acoustic signals in vortex current fields using numerical methods and experimentally verify their accuracy. Then, an acoustic signal-based monitoring method is proposed for vortex current fields near islands, thereby providing a new theoretical basis and technical support for the construction of near-shore waters engineering and navigation safety. Although the simulation method is numerical, it is necessary for future application in real marine environments. The paper is structured as follows: Section 2 introduces the data and methods, including the HYCOM marine dataset of the vortex current field near Zhoushan Islands, the numerical simulation method for acoustic signals based on ray acoustics, the TRM for extracting vortex acoustic signal phases, and the conditions for simulating the vortex current field. Section 3 verifies the accuracy of the numerical simulation method by comparing experimental and simulation results. Section 4 presents the monitoring results for the Barley Straw Reef vortex current field and the analysis of the performance of the monitoring algorithm under different conditions. Finally, Section 5 summarizes the main conclusions and suggestions for future work.

2. Materials and Methods

2.1. HYCOM Data

The numerical model of HYCOM (Hybrid Coordinate Ocean Model) is used for the study of the marine environment and plays a crucial role in global ocean research [19]. In this paper, we have chosen the HYCOM+NCODA Global 1/12 Analysis (GLBa0.08) data version (https://tds.hycom.org/thredds/catalogs/GLBv0.08/expt_93.0.html (accessed on 12 April 2024)) from its experimental versions to obtain ocean current velocity data. The data are gridded using the anatomic Mercator-curvilinear HYCOM horizontal grid with a spatial resolution of 1/12° × 1/12°. As shown in Figure 1, a significant vortex field exists in the northern part of Zhoushan Island, Liangheng Island, near Barley Straw Reef, where the flow field is complex and the velocity is high. This vortex field is taken as an example, and the radial basis function is used to obtain its continuously varying velocity distribution using HYCOM model oceanographic data.

2.2. Principles and Numerical Simulation of Vortex Current Field Monitoring

2.2.1. Principle of Vortex Current Field Monitoring

The propagation of acoustic waves in a vortex field is accompanied by changes in amplitude and phase due to the motion of the vortex field. In Figure 2, the vortex field is regarded as a variable speed medium; in the countercurrent region, the direction of acoustic wave propagation is opposite to the direction of vortex medium movement, the speed of acoustic propagation decreases, the intensity of acoustic signals is enhanced, and the phase lag and the downstream region are opposite to it. The deformation of the acoustic wavefront caused by the vortex field carries information about phase changes that can reflect the morphological characteristics of the vortex field. This provides a theoretical basis for the monitoring of vortex fields.

From the study in [12], it is known that the phase difference curve of the received acoustic signal exhibits a typical and distinctive Z-shaped pattern. The maximum change in the phase difference curve is referred to as the phase jump variable. This curve is closely linked to the parameters of the vortex field, which enables the construction of a mapping relationship between the phase of the received acoustic signal and the characteristic parameters of the vortex field. The vortex field parameters can be determined by analyzing the phase difference curve of the received acoustic signal, including the position of the vortex core and its characteristic radius.

2.2.2. Ray Acoustic Modeling in Vortex Field

The ray acoustic method is commonly used to calculate the acoustic fields, which has been proven to be highly effective in studying the characteristics of sound propagation in vortex fields [20,21,22]. As shown in Figure 3, let ${\mathit{w}}_Q$ be a moving point on a wavefront $t=\tau (\mathit{x})$ with velocity $c\mathit{n}$, where $\mathit{n}$ refers to a unit vector perpendicular to the wavefront, and when the vortex medium is moving with velocity $v$, the velocity of point ${\mathit{w}}_Q$ on the wavefront is expressed as follows:

$$\mathit{u}=c\mathit{n}+\mathit{v}$$

(1)

where it is evident that in a vortex field, the direction of the ray is typically not perpendicular to the wavefront. The variable $\alpha$ represents the angle between the direction of ray propagation and the normal of the wavefront.

The set of differential equations governing the behavior of rays in a vortex field is expressed as follows [23]:

$$\{\begin{array}{c}\frac{\mathrm{d}\mathit{w}}{\mathrm{d}t}=c\mathit{n}+\mathit{v}\\ \frac{\mathrm{d}\mathit{s}}{\mathrm{d}t}=-\frac{(1-\mathit{s}·\mathit{v})\nabla c}{c}-\mathit{s}\times (\nabla \times \mathit{v})-(\mathit{s}·\nabla )\mathit{v}\end{array}$$

(2)

where $\mathit{s}$ refers to the slowness vector, $\mathit{w}$ represents the acoustic trajectory, $\mathit{v}$ denotes the rotational velocity of the vortex medium, and c stands for the sound velocity distribution of the vortex field. The vortex field $\gamma =\nabla \times \mathit{v}$ is called vorticity, which is an important physical quantity for reflecting the rotational characteristics of the vortex field. The ray trajectory in the vortex field is obtained by solving Equation (2) using the Lungkuta method.

The sound pressure amplitude of the ray is determined by applying the law of the conservation of acoustic energy [24]:

$$\frac{P^2\left|\mathit{u}\right|S}{\rho c^2(1-\mathit{s}·\mathit{v})}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

(3)

For a 2D vortex field, assuming that refraction only occurs in the xoy plane, the cross-sectional area S at the reception point can be calculated from the paths of the two rays with a small difference in the initial grazing angle $\mathsf{\Delta }\theta$ (typically 0.0005–0.01 rad).

The cross-sectional area of the ray tube changes when refraction occurs in a vortex field with a density $\rho$ and speed of sound c, as depicted in Figure 4:

$$S_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}=ld\mathsf{\Delta }\eta$$

(4)

The cross-sectional area of a ray tube is determined by the absence of refraction in a homogeneous stationary medium with a specific density ${\rho }^{\prime }$ and speed of sound $c^{\prime }$:

$$S_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}=l^{\prime }{d}^{\prime }\mathsf{\Delta }\eta$$

(5)

The area corresponding to the ray tube is not perpendicular to the wavefront surface because the direction of sound propagation is not directly aligned with the wavefront:

$$S_{\mathrm{r}\mathrm{a}\mathrm{y}}=S_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}\cos \alpha$$

(6)

where $\alpha$ refers to the angle between the ray propagation direction and the normal to the wavefront surface; hence, $\cos \alpha =\left[\mathit{u}·\left(\mathit{u}-\mathit{v}\right)\right]/c\left|\mathit{u}\right|$.

Substituting this into Equation (3), the vortex field ray amplitude is expressed as follows:

$$P^2=\frac{P_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}^2{l^{\prime }}^2\mathsf{\Delta }\theta (1-\mathit{s}·\mathit{v})\rho c^4}{ld\mathit{u}·\left(\mathit{u}-\mathit{v}\right){\rho }^{\prime }{c^{\prime }}^2}$$

(7)

where P_static refers to the amplitude of the ray in the stationary medium, $d=\sqrt{{\left(x_{Q_1}-x_{Q_2}\right)}^2+{\left(y_{Q_1}-y_{Q_2}\right)}^2}$.

The propagation time $\tau$ of the ray from the distance l traveled by $\mathit{w}$ is found:

$$\tau ={\displaystyle \underset{l}{\int }\frac{1}{c+\mathit{v}·\mathit{n}}}\mathrm{d}l$$

(8)

This gives the phase of the propagation of the ray $\phi$:

$$\phi =2\pi f\tau$$

(9)

The sound pressure signal for each ray can be obtained using Equations (7) and (9). According to ray theory, the sound field at the receiving point is the result of the contribution of each intrinsic ray. In other words, the sound pressure at the receiving point is determined by the combination of intrinsic rays. By superimposing the intrinsic rays that arrive at the receiving point in the vortex field, we can obtain information about the amplitude and phase of the sound pressure. This enables the numerical simulation of the acoustic signal at the receiving point.

2.2.3. TRM Technique to Extract the Phase of Vortex Acoustic Signals

The TRM technique has become a research hotspot in hydroacoustic technology in recent years due to its ability to achieve spatial and temporal adaptive focusing of time-reversal signals [25,26].

As shown in Figure 5, an emitted acoustic signal at a certain frequency is used as the sound source, which is emitted by TRM1 with a plane wave with a certain power:

$${p_i}^1(t)=P(i)\exp (2\pi ft)$$

(10)

When the acoustic waves pass through the vortex field, due to the action of the vortex field on the speed of sound propagation, the acoustic signals at different locations take different propagation times ${t_i}^{1\to 2}$ when they reach TRM2 from TRM1.

$${S_i}^{1\to 2}(t)=P{(i)}^{1\to 2}\exp (2\pi f(t-{t_i}^{1\to 2}))$$

(11)

TRM2 processes the received signal based on time-reversal: the target information is obtained by adding window extraction to the time domain of the received signal, and the signal is sent back to the vortex field after reversing the order in the time domain. Then, the processed signal is expressed as follows:

$${p_i}^2(t)={S_i}^{1\to 2}(-t)=P{(i)}^{1\to 2}\exp (2\pi f(T+{t_i}^{1\to 2}-t))$$

(12)

When the time-reversal signal passes through the vortex field again, the propagation time required to reach TRM1 from TRM2 becomes ${t_i}^{2\to 1}$. The acoustic signal received by TRM1 is then expressed as follows:

$${S_i}^{2\to 1}(t)=P{(i)}^{2\to 1}\exp (2\pi f(T+{t_i}^{1\to 2}-t+{t_i}^{2\to 1}))$$

(13)

where ${S_i}^{1\to 2}(t)$ refers to the signal received by each array element i of TRM2 when TRM1 is the transmitter and ${S_i}^{2\to 1}(t)$ denotes the signal received by each array element i of TRM1 when TRM2 is the transmitter.

Further iterative operations are performed to numerically simulate the received signals from the TRM in the presence and absence of the vortex, respectively, to obtain the amplitude ratios and phase differences of the vortex acoustic signals received by the TRM for different numbers of time-reversals.

2.2.4. Vortex Field Simulation Conditions

During the numerical simulation, the acoustic frequency of the sound source was set to f = 35 KHz. As shown in Figure 6, 60-element sensors were arranged at equal spacing at both the TRM1 and TRM2 ends. These sensors transmitted planar acoustic waves and received acoustic signals passing through the vortex field. The array elements of the transmitting and receiving arrays were uniformly distributed at a spacing of 1 m. The horizontal coordinate of the TRM1 array was set to x = 0 cm, and the vortex nucleus characteristic radius was denoted by r. The spacing of the TRMs array was D = 200 m, and the ambient sound velocity in the water was set to c = 1500 m/s.

The numerical simulation of the acoustic signal is performed in two vortex scenarios, with the amplitude ratios $P_i^R$ and phase differences $\mathsf{\Delta }{\phi }_i$ at the reception points calculated as follows:

$$\begin{array}{l}{P}_i^R=\frac{P_i^0}{P_i}\\ \mathsf{\Delta }{\phi }_i={\phi }_i-{{\phi }_i}^0\end{array}$$

(14)

3. Accuracy Validation of Acoustic Signal Simulation Methods

To validate the accuracy of the numerical simulation method for acoustic signals, as described in Section 2, in this paper, the acoustic signals are experimentally measured in a laboratory setting. The focus of the experiment is the vertical axis vortex flow field, as depicted in Figure 7. The experiment was conducted in an open circulating water tank divided into two sections, namely the water storage area and the vortex area, with a water-stable baffle serving as the dividing line.

The experimental test device, as shown in Figure 7b, includes an open circulating water tank, a water-stable baffle, a signal generator, a ternary ultrasonic sensor array, and a data acquisition instrument. First, the water outlet hole B is opened to allow circulating water to flow through the water inlet A into the water storage area. The water then passes through the water-stable baffle to stabilize the flow and reduce fluctuations before entering the vortex area. Subsequently, the inflow flow rate is adjusted to establish a stable vortex in this region. The description of the experimental apparatus and equipment is shown in

Table 1. Model and parameters of experimental instruments and equipment.

Experimental Equipment	Model	Parameters
Circulating water tank	Customizable	500 mm × 400 mm × 400 mm. Outlet diameter of 2 cm
Water-stable baffle	Customizable	Inlet opening diameter of 2 cm
Signal generator	DH-1301 (Jiangsu Donghua Testing Technology Co., Taizhou, China)	Logarithmic sweep: 1.0–2.0 oct/s
Ultrasonic sensor	500E35TR-1 (ShenZhen OSENON Technology Co., Ltd., Shenzhen, China)	Nominal frequency 500 KHz, beamwidth 6°
Data acquisition instrument	USB8582 (Beijing ART Technology Development Co., Beijing, China)	8 channels with 12-Bit ADC resolution

.

Odgaard [27] derived a formula for the distribution of vertical axis vortex velocity using experimental data.

$$v(r)=\frac{\mathsf{\Gamma }}{2\pi r}\left(1-\exp \left(-1.25\frac{r^2}{{r_0}^2}\right)\right)$$

(15)

In the given equation, the characteristic radius r₀ of the vortex nucleus in the experiment is equivalent to the diameter of the water outlet B. Therefore, it can be stated that r₀ = 2 cm, Mach number $Ma=5.9\times {10}^{-4}$, and vorticity $\Gamma =2.8\pi r_{0}cMa=0.157 {\mathrm{m}}^2/\mathrm{s}$.

As measured in the experiment, the vortex axis of the vertical axis vortex was observed to be close to the plumb direction, resulting in a more stable vortex motion and a symmetrical morphology. The sound speed in water was c = 1500 m/s. The ultrasonic sensor emitted a single-frequency sinusoidal signal at an acoustic signal emission frequency of f₀ = 500 KHz and a sampling frequency f_s = 100 MHz. Figure 7a shows the established coordinate system with the water outlet B as the origin and the receiving transducer G in the middle as the reference coordinate for the receiving array. The transmitting array remained stationary, while the receiving array moved horizontally by changing the position of sensor G. Five sets of experimental conditions were set up, with sensor G moving at intervals of 1 cm, and the total movement of $\mathsf{\Delta }{x}_i=-5+i$ where i = 0, 1, 3, …, 10 cm. The acoustic signals were measured using the receiving sensor at 11 positions with and without vortex, and the measurement was repeated five times at each position.

The experimental data and numerical simulation results of acoustic signals were analyzed and processed under various working conditions. The results of the amplitude ratio and phase difference of the acoustic signals at different receiving points are shown in Figure 8. It can be observed that the numerical simulation results of the acoustic signals passing through the vortex field and the vortex-free field are consistent with the experimental results. The root-mean-square error of the amplitude ratio is 2.81%, and the relative error of the phase difference is 1.3%. Comparing these results with the experimental data verified the accuracy of the acoustic signal simulation method proposed in this paper and, consequently, affirmed the credibility of the numerical simulation method.

4. Results and Discussion

4.1. Barley Straw Reef Vortex Field Monitoring Results

4.1.1. Velocity Distribution of the Vortex Field in the Barley Straw Reef

Using the HYCOM marine ambient velocity as a sample point, the velocity distribution of the Barley Straw Reef vortex field can be obtained based on radial basis function interpolation. The coordinates of the vortex core center are set to be (0,0) and (0,10) for simulation conditions I and II after the normalization of the vortex core characteristic radius, and the vortex core characteristic radius is 5 m and 10 m, respectively. Figure 9 shows the tangential velocities and velocity distributions with different radii originating from the vortex core center under both simulation conditions (the coordinates are normalized by the vortex characteristic radius). A comparison was made between the interpolated vortex field velocity values, as shown in Figure 9a,c, and the HYCOM data values, which reveals that the radial basis interpolation function can accurately capture the vortex field velocity distribution.

4.1.2. Characteristic Parameters of the Barley Straw Reef Vortex Field

In Section 2 of this paper, the amplitude and phase of the acoustic signal in the vortex field are determined based on the ray acoustic model. This model is critical for simulating the acoustic signal in the vortex field. The acoustic signals passing through the vortex field of the Barley Straw Reef vortex field under both conditions were then simulated numerically to obtain the phase difference curves at various positions. The results are presented in Figure 10.

According to the study in [12], the zero-phase point in the phase difference curve of the vortex field is typically near the center of the vortex nucleus. Additionally, the reception point reflecting the peak point of the phase difference corresponds to the characteristic radius of the vortex nucleus. Therefore, it is feasible to determine both the center and characteristic radius of the vortex nucleus based on the phase difference curve of the acoustic signal. The inversion characteristic parameters obtained under the two vortex field conditions at Barley Straw Reef are shown in

Table 2. Barley Straw Reef vortex field monitoring results (The locations of the vortex core center are normalized by the vortex characteristic radius).

Vortex’s Current Field Condition	Characteristic Parameters of the Vortex Field	Simulation Parameters	Estimated Value	Relative Error
I	Location of vortex core center	y = 0	y = 0	0%
	Characteristic radius of the vortex nucleus	5 m	5.32 m	6.4%
II	Location of vortex core center	y = 1	y = 1	0%
	Characteristic radius of the vortex nucleus	10 m	9.17 m	8.1%

.

The analysis of the data in Table 2 showed that the relative errors of the vortex core center position and characteristic radius were both within 10% under both vortex field simulation conditions. This proves that the monitoring method proposed in this paper can accurately capture the dynamic changes of the vortex field at Barley Straw Reef.

4.2. Extraction of Vortex Field Phase Features using the TRM Technique

4.2.1. Numerical Simulation of Vortex Acoustic Coupling Based on TRM

The phase difference of the vortex acoustic signals received by the TRMs with different numbers of simultaneous inversions is numerically simulated under simulation condition I, as described in Section 4.1.1 as an example. The results are shown in Figure 11, where N denotes the number of times the acoustic signals pass through the vortex field.

As can be seen from

Table 3. Phase jumps of acoustic signals with different numbers of simultaneous inversions.

Inverse Time Count	N = 1	N = 2	N = 3	N = 4	N = 5
Phase jump (rad)	0.1944	0.4214	0.6033	0.8328	1.0346

, after the acoustic signal passes through the vortex field N times, that is, after N − 1 time inversion processes, the phase jump $\mathsf{\Delta }{\varphi }_N^P$ is $\mathsf{\Delta }{\varphi }_N^P=N\ast \mathsf{\Delta }{\varphi }_1^P$. The TRMs array can amplify the phase change of acoustic signals passing through the vortex field, which can be approximated as a linear vorticity amplifier.

The marine environment brings noise interference and restricts the infinite increase in the number of time-reversal N. Specifically, in the time-reversal process, assuming that the noise in each time-reversal is independent of each other and the power is $P_N$, the signal power after the Nth time-reversal is $P_S^N$, then the noise power after the Nth time-reversal is $P_{noise}^N=\sqrt{N}·P_N$, which indicates that with the increase in the number of times of time-reversal N, the phase hopping variable will be enlarged N times, but the noise power will be enlarged $\sqrt{N}$ times of the original with it. Therefore, it is important to select an appropriate number of time-reversals to avoid the over-amplification of the noise and affecting the extraction of the acoustic signal phase in the vortex current field. The simulation shows that a linear amplification relationship can be maintained when N is less than 7, which also satisfies the conclusions obtained from the study in [17]. For calculational convenience, N = 5 will be used.

4.2.2. Numerical Simulation of Vortex Field Acoustic Signals under Different Signal-to-Noise Ratio Conditions

In general, the Mach number in water is low, and the phase distortion generated by the vortex field is very weak. In marine environments with vortex fields, the propagation of acoustic waves depends on various factors including intrinsic noise. The other factors include phase perturbations caused by undulations in the acoustic field, seawater interface, and changes in seawater density. The level of phase perturbation is stochastic, which cannot be linearly superimposed. However, if the perturbation is too large, it will obscure the phase aberration caused by the vortex current medium. The TRM technique can be used to effectively amplify the phase jumps caused by the vortex current field, making it useful for extracting these jumps even in the presence of other phase perturbation factors.

Assuming that the phase perturbation caused by other marine environments follows a Gaussian white noise distribution, in this paper, the simulation condition I described in Section 4.1.1 is taken as an example. In the numerical simulation, random phase perturbation quantities with varying variances are added. Time-reversal simulations are then conducted five times to investigate the impact of these phase perturbation quantities on the performance of the vortex field monitoring method proposed in this paper. Figure 12 shows the phase difference curves of the vortex acoustic signals for three time-reversals (N = 1, N = 3, and N = 5) at different phase perturbation quantities.

Upon phase perturbations with varying variance sizes, the mean value of the phase difference curve was obtained through a smoothing process. The resulting phase jump volume and error of the vortex-acoustic signal at N = 1 and N = 5 are presented in

Table 4. Phase jumps and errors of vortex acoustic signals at different phase perturbation quantities.

Phase Perturbation Variance	N = 1 Phase Jump Volume	Relative Error	N = 5 Phase Jump Volume	Relative Error
No perturbation	0.1944		1.0346
2.05 × 10−4	0.1943	0.0514%	1.0351	0.0676%
8.33 × 10−4	0.1944	0%	1.0321	0.2223%
0.0032	0.1946	0.1028%	1.0337	0.0676%

, respectively. As can be seen, the phase difference hopping error is less than 1% after data processing, regardless of the number of simultaneous inversions. Furthermore, the N-fold linear relationship is maintained. The smoothing process reduces the random error caused by disturbances to a certain extent, but the phase hopping variance caused by vortex current motion remains linearly larger after time inversion. Therefore, in marine environments with random disturbances, the TRM technique facilitates the extraction of the phase of the acoustic signal from the vortex field compared to the case without time-reversal.

4.2.3. Impact of Algorithm Performance with Different Signal-to-Noise Ratios

The TRM amplifies the phase change of the vortex field acoustic signal, resulting in an increase in the time delay magnitude after multiple time-reversals. For instance, in the simulation condition I described in Section 4.1.1, the time delay τ is approximately 10⁻⁶ with a time-reversal number of N = 1. However, after N = 5 time-reversals, the time delay τ increases to approximately 10⁻⁵. To evaluate the accuracy of the time delay estimation algorithm, the delay data errors are calculated at different signal-to-noise ratios for N = 1 and N = 5 time-reversals, as shown in Figure 13.

Under the same SNR condition, the magnitude of time delay has less effect on the accuracy. The reason is that the accuracy of time delay depends mainly on the SNR. As shown in Figure 13, the time delay accuracy decreases significantly when the signal-to-noise ratio is lower than 10 dB, regardless of the time delay magnitude. Therefore, to ensure the accuracy of the proposed acoustic phase feature extraction in this paper, it is crucial to maintain a certain signal-to-noise ratio at the receiving end. Additionally, the time-reversal processing mainly amplifies the magnitude of time delay, making it easier to detect the required acoustic phase changes caused by vortex field perturbations. This process has minimal impact on the accuracy of time delay measurement.

5. Conclusions

In this paper, a method for monitoring vortex current fields near islands based on acoustic signals was proposed. This method is significant for improving the stability of marine engineering construction and enhancing the safety of maritime navigation. In this paper, the phase feature extraction of the acoustic signals passing through the vortex current field of Barley Straw Reef was used, based on the HYCOM marine dataset, to monitor and forecast the vortex current field. In addition, the performance of the vortex current field monitoring algorithm under different conditions was also investigated. The following conclusions are drawn:

(1)

In this paper, an equation was derived to calculate the acoustic pressure amplitude based on ray acoustics, and a numerical simulation method was developed for acoustic signals in vortex current fields. In addition, the accuracy of this simulation method was verified by comparing it with experimental results.

(2)

It was found that the TRM technique can linearly amplify the phase jump generated by the vortex current field and, based on the time-reversal effect, can extract the phase difference information of acoustic signals generated by vortex currents in complex marine environments.

(3)

Based on the results of the numerical simulation of acoustic signals, the study is able to determine the vortex core center and characteristic radius of the vortex field of Barley Straw Reef, which initially verifies the feasibility of using acoustic signals to monitor near-shore vortex fields.

In this paper, several issues were identified in using the acoustic signal method to monitor the vortex current field of near-shore islands and reefs. These issues can serve as a guide for future research by the follow-up group.

Firstly, the numerical simulation failed to fully reflect the real marine environment, and it is necessary to expand the scope of work to include the real 3D vortex current field in the marine environment and consider the impact of marine boundary conditions.

Secondly, while this paper takes into account the influence of phase perturbation caused by the marine environment in addition to the vortex current field, it simplistically assumes that the perturbation follows a smooth Gaussian distribution. Therefore, further study is needed to fully understand its statistical characteristics.

Lastly, the use of acoustic signals still faces many uncertainties, particularly since the technique has not been tested in a real marine environment, which lacks further verification and improvement through experiments.