Modulation of Deep-Sea Acoustic Field Interference Patterns by Mesoscale Eddies

Abstract

The interference structure of the underwater acoustic field has significant theoretical and engineering application value in underwater target detection and marine environmental monitoring. The impact of mesoscale eddies on acoustic interference striations remains a significant research gap. Through simulation experiments, the influence of the eddy propagation process on the interference striations of the sound field was studied and analyzed. The first few orders of the interference striations in the spatial domain are more susceptible to the influence of eddies. As a warm eddy gradually propagates away from the source, there is a shift of interference striations in the range–frequency domain toward lower frequencies, while the cold eddy does the opposite. Meanwhile, the arrival structure at a fixed point process undergoes regular changes.

1. Introduction

Mesoscale eddies are ubiquitous dynamical phenomena in the ocean. The relative independence of water masses captured by mesoscale eddies results in marked differences in the physical properties of seawater inside and outside the eddies [1,2], which has a significant impact on underwater acoustic propagation. Research on the acoustic effects of eddies first began in the 1970s. Vastano and Owens [3] revealed the acoustic field perturbation caused by the cyclonic eddy in 1967 for the first time in the Sargasso Sea, where the acoustic environment is relatively homogeneous. And a dispersion phenomenon of the ray path in the area where the sound channel depth changed was found with the ray computations. Weinberg and Zabalgogeazcoa [4] further verified the modulation of sound propagation by eddies. In the same period, it was found that the coupling of acoustic energy between the subsurface channel and the deep ocean channel is frequency-dependent through experiments in the Eastern Australian Sea [5,6].

In the absence of the synchronous survey between the eddy and acoustic, researchers have developed a series of parametric acoustic models to quantify the effect of the eddy features on acoustic propagation [7,8,9]. Baer [10] combined the split-step parabolic equation with the Henrick model, which revealed a phenomenon where the gain of the signal received by the vertical array increased due to eddy, and it was observed that there was a horizontal propagation angular excursion of more than 0.5°. Since the 1980s, the North Pacific Acoustic Laboratory (NPAL) has conducted a series of experiments, such as SLICE89, Acoustic Engineering Test, Alternate Source Test, ATOC, PhiSea09 and OBSAPS, systematically investigating the mechanism of marine environmental variability on sound propagation [11,12,13]. Among them, the PhiSea10 experiment confirms that strong eddy activity can lead to significant ups and downs in sound propagation time [14], while an acoustic experiment in the East China Sea found that cold eddies have observable modulation features on the modal structure, group velocity and multipath structure [15].

Acoustic technology and its capabilities deserve particular emphasis [16]. In the field of inversion of marine environmental parameters, it was observed that measurements using acoustic means can overcome the limitations of point measurements and better characterize the exchange of water by small-scale eddies [17]. Porter and Spindel [18] provided the first motion-corrected low-frequency acoustic fluctuation spectra, demonstrating linear frequency scaling of phase and distinguishing drifting versus moored receiver regimes. Virovlyansky et al. [19] quantified mesoscale-induced modal coupling and timefront distortion via ray action variables, showing that accumulated internal wave effects dominate at long ranges. Cornuelle et al. [20] investigated the capability of ocean acoustic tomography enhanced by a shipborne receiver for mapping the mesoscale ocean sound speed field. Acoustic refraction caused by mesoscale ocean processes can significantly affect the accuracy of the remote localization of low-frequency acoustic sources. This challenge is particularly prominent in the mesoscale energy-active regions of the ocean [21,22].

The image interference effect has been central to investigations in underwater acoustics and sonar [23] because it has rich practical applications in passive localization of acoustic targets [24,25], signal processing [26] and inversion of marine environment parameters [27]. Guthrie et al. [28] observed the interference structure of a real ocean waveguide through deep-water acoustic propagation experiments off Antigua. Subsequent theoretical breakthroughs emerged with the conceptualization of waveguide invariants, providing a unified framework to quantify interference characteristics across varying propagation conditions. Rouseff [29] systematically investigated shallow water internal wave impacts on waveguide invariant distributions. A method for estimating waveguide invariants using the two-dimensional Fourier transform was proposed [26], and the relevant theory was used for source motion compensation [30]. Through shallow water summer sound speed profile (SSP) measurements, Turgut et al. [31] empirically demonstrated the depth-dependent behavior of waveguide invariants. Emmetière et al. [32] employed normal mode theory to elucidate the mechanisms of deep sea interference structure formation, while Weng et al. [33] derived specific analytical approximations for interference patterns in different regions of the deep sea acoustic field. Traditional research on pattern formation mechanisms has primarily depended on normal mode theory. However, an alternative theoretical framework attributes the striation evolution characteristics to the behavior of the impulse response [34]. Given the widespread occurrence of mesoscale phenomena in ocean dynamics and the operational significance of interference structures in applications, it is surprising that systematic investigations into how mesoscale eddies impact deep sea acoustic interference structures are still lacking. This represents a significant gap in underwater acoustics knowledge. Advancing quantitative studies on eddy–interference interactions is essential. Understanding the modulation of interference striations by mesoscale eddies can help to improve inversion algorithms for ocean parameter estimation and enhance the robustness of the sonar system in an eddy-rich environment. This study simulates the changes in interference patterns caused by eddies and analyzes the underlying mechanisms using ray tracing theory and arrival structure. It establishes, for the first time, a quantitative modulation relationship between mesoscale eddies and acoustic interference striations, revealing a distinct phenomenon whereby eddy-induced sound speed anomalies alter path differences to drive energy redistribution in interference patterns.

2. Materials and Methods

2.1. Model Setup

The ocean is a complex and dynamic system. More than $75\%$ of mesoscale eddies with lifespans of ≥16 weeks exhibit westward propagation, primarily driven by the $\beta$-effect (planetary vorticity gradient) and advection by background currents. Only about $25\%$ of eddies show eastward movement, concentrated in strong eastward-flowing current regions such as the Antarctic Circumpolar Current, the Gulf Stream and its extension and the Kuroshio–Oyashio confluence zone [35,36,37]. These eastward-moving eddies, advectively driven by strong background currents, typically have shorter propagation distances (<1800 km) and shorter lifespans. Eddy propagation speeds approximate the phase speed of long Rossby waves but exhibit significant latitudinal dependence: faster in low-latitude regions and slower at high latitudes. Notably, eddies in the Southern Hemisphere move $20\%$ faster than those in the Northern Hemisphere, though the underlying mechanisms remain unclear [35]. Due to the movement of mesoscale eddies, synchronous surveys of the moving eddies on the acoustic sensing methods are quite difficult and thus have been rarely conducted. Thus, this paper has conducted several simulations to investigate the characteristics of the acoustic fields due to eddies. From the perspective of acoustic studies, the distribution of the sound speed due to the existence of eddies can be parameterized by Gaussian eddy models. It assumes that the spatial distribution of the sound speed consists of a linear superposition of the climatic state $c_0\left(z\right)$ and the eddy-induced sound speed perturbations $\delta c(r,z)$:

$$c(r,z)=c_0\left(z\right)+\delta c(r,z)$$

(1)

In the deep sea, the climatic state of the sound speed profile (SSP) is described as a Munk profile [38], and its general form is:

$$c_0\left(z\right)=1500[1+\epsilon (e^{-\eta }-(1-\eta )\left)\right]$$

(2)

where $\eta =2\left(z-z_{min}\right)/B$. The axis depth $z_{min}$ defines the depth of minimum sound speed, taken as 1100 m. The depth scale parameter B is set to 1100 m, analogous to the thermocline e-folding depth. $\epsilon$ is a dimensionless parameter representing the magnitude of the sound speed change, taken as 0.0057 [39,40,41]. The variation of sound speed with depth is shown in Figure 1a.

The coordinate system is illustrated by Figure 2. The coordinate origin O is at the sea surface above the source. The z-axis points downward from the sea surface, and the r-axis extends horizontally toward the receiver. The eddy-induced sound speed perturbation can be given by [7,41]:

$$\delta c(r,z)=\mu e^{-{\left({\displaystyle \frac{r-L}{R}}\right)}^2-{\left({\displaystyle \frac{z-D}{Z}}\right)}^2}$$

(3)

where $\mu$ is the eddy intensity, indicating the maximum sound speed disturbance caused by the eddy and taking a negative (positive) value for a warm (cold) eddy. L denotes the horizontal position of the eddy center and D denotes the vertical coordinate of the eddy center. r represents the horizontal distance from the eddy center, and z denotes the vertical distance from the eddy center. Figure 1b,c show the distribution of sound speed induced by a warm ($\mu$ = 10 m/s) and cold ($\mu$ = −10 m/s) eddy. The eddy core maintains a constant depth of 300 m, with horizontal and vertical scales of 60 km and 800 m, respectively. Oceanic eddies in real-world marine environments are perpetually in motion. To simulate eddy propagation, the horizontal coordinate L is treated as time-dependent, denoted as $L\left(t\right)$, where t represents time.

The experiment scenarios for simulations in latter sections are shown in Figure 2. Consider an eddy that moves at a constant speed in a straight line, and all the other properties of the eddy remain unchanged during the movement. Figure 2a shows the dynamic movement of the eddy, where the acoustic source is fixed at (0 km, 100 m). Initially, the center of a mesoscale eddy is positioned at (−70 km, 300 m). As the eddy propagates towards the source over the time, its center passes through the sound source eventually. The specific moment is illustrated in Figure 2b, where the eddy center lies directly beneath the sound source at position (0 km, 300 m). Then the eddy continues to move forward, with its center gradually moving away from the source location and reaching the range of 70 km. The sound speed profile is taken as a Munk profile [38]. The geoacoustic parameters of the seafloor are shown in Figure 2a,b. The study area is defined in the region to the right of the y-axis, with the hydroacoustic receiver assumed to be located in the first quadrant.

2.2. Theory

The multipath effect is important in the ocean and commonly observed in the acoustic array. At deep water, when the source and the receiver are within the direct arrival zone (DAZ), bright and dark striations due to the interference of the direct (D) and the surface-refracted (SR) rays are shown, which is known as Lloyd’s mirror effect (LME). This effect is depicted in Figure 3, where S and $S^{\prime }$ represent the true and the image acoustic sources. The source and receiver are at depths $z_s$ and $z_r$, respectively. The horizontal range is r. $R_1={\left(r^2+{\left(z_s-z_r\right)}^2\right)}^{1/2}$ and $R_1={\left(r^2+{\left(z_s+z_r\right)}^2\right)}^{1/2}$ denote the ray lengths (or the slant ranges) of the D and SR arrivals. The total acoustic field $p(r,z)$ is considered as the sum of the contributions from the two sources S and $S^{\prime }$, given by [42]

$$\begin{array}{c}p(r,z,\omega )=S\left(\omega \right)\left(e^{ik{R}_1}/R_1-e^{ik{R}_2}/R_2\right)\end{array}$$

(4)

where $S\left(\omega \right)$ is the complex source spectrum at angular frequency w, $k=w/c$ is the wave number and c is the reference sound speed. As shown in Figure 3, for a sufficient deep array and a shallow source in the direct arrival zone (DAZ), $R_1\approx R_2\approx R$, one obtains

$$\begin{array}{cc}\hfill p(r,z,\omega )& \approx {\displaystyle \frac{S\left(\omega \right)}{R}}\left(e^{ik{R}_1}-e^{ik{R}_2}\right)\hfill \\ \hfill & ={\displaystyle \frac{S\left(\omega \right)}{R}}\left(e^{i\omega t^D}-e^{i\omega t^{SR}}\right)\hfill \\ \hfill & =-{\displaystyle \frac{2S\left(\omega \right)i}{R}}{e}^{i\omega (t^D+{\tau }_0/2)}sin(\omega {\tau }_0/2)\hfill \end{array}$$

(5)

where $t^D=R_1/c$ and $t^{SR}=R_2/c$ are the arrive times of the D and $SR$ paths and ${\tau }_0=t^D-t^{SR}$ is the D–SR time delay. The acoustic intensity is given by

$$\begin{array}{cc}\hfill I_P\left(r,z,\omega \right)& ={\left|p\left(r,z,\omega \right)\right|}^2/\rho c\hfill \\ \hfill & \approx 4{\left|S\left(\omega \right)\right|}^2{sin}^2\left(\omega {\tau }_0/2\right)/\rho c{R}^2\hfill \\ \hfill & =2{\left|S\left(\omega \right)\right|}^2\left[1-cos\left(\omega {\tau }_0\left(r,z\right)\right)\right]/\rho c{R}^2\hfill \end{array}$$

(6)

From the above equation, it can be found that the acoustic intensity of the pressure field in the DAZ has very similar interference patterns modulated by ${sin}^2\left(\omega {\tau }_0/2\right)$, which shows constructive and destructive interference striation (CIS and DIS) patterns. According to Equation (6), CISs and DISs correspond to the maximums and minimums of the acoustic intensities, which yield $cos\left(\omega \tau \right(r,z\left)\right)=-1$ and $cos\left(\omega \tau \right(r,z\left)\right)=1$, respectively. Then, given the frequency and the source depth, the nth CIS curve satisfies the relationship given by

$$\omega {\tau }^{CIS}(r_n,z_r\left)=\right(2n-1)\pi ,n=1,2,\dots$$

(7)

For the nth DIS curve, it satisfies

$$\omega {\tau }^{DIS}(r_n,z_r)=2n\pi ,n=1,2,\dots$$

(8)

Since the CIS and DIS striation patterns can be used for the source depth estimation using vertical linear arrays or acoustic vector sensor arrays, a lot of work has been performed to predict them [43]. When eddies exist and propagate near the acoustic array in deep water, the interference striation patterns will change due to the perturbation of the SSP caused by eddies. The following section will investigate the impact of the existence of eddies on the characteristics of the acoustic field generated by a narrowband source in the spatial domain and a broadband source in the spatial frequency domain.

3. Simulations and Discussions

3.1. Spatial Domain

Usually, in underwater acoustics, the transmission loss (TL) is used to show the intensity of the acoustic field. It is defined as the ratio of sound intensity $I_P\left(r,z,\omega \right)$ at a field point to the intensity $I_0$ at a reference distance from the source (usually 1 m, $I_0\approx 6.50\times$ 10⁻¹⁹ W/m²), expressed in decibels (dB): $TL=-10log(I_P\left(r,z,\omega \right)/I_0)$. The ray model Bellhop is used here to model the acoustic field as shown in Figure 4 [44,45], where the acoustic source of its frequency 100 Hz is at the range of 0 km and the depth of 100 m. Figure 4a shows the TL without eddies, while Figure 4b,c represent the TLs of the cold and warm eddies, respectively, whose centers are directly below the sound source. Figure 4 shows that bright and dark striations appear within the DAZ due to the LME, with CISs and DISs overlaid as while dotted lines in the TLs. The theoretical CISs and DISs are predicted by Equation (7) using the D–SR time delay difference obtained from Bellhop under the given environment parameters and the source array parameters [44,45]. The CISs are assigned labels of n = 1, 2, …, 13 corresponding to Equation (7), which correspond to progressively large launching angles. The time delay for each CIS is also shown in Figure 4. The theoretical striations of different orders match well with the CISs of the TL results.

Moreover, from Figure 4a, the arrival times of the D and SR rays with large launching angles will not change with the SSP within the range of 2H (H for water depth). As the source-receive range increases to the boundary of the DAZ, arrival times of the D and SR rays will be greatly affected by SSP. Thus, as shown in Figure 4b,c, only positions of first to third CISs change when eddies exist. Since first to third CISs correspond to small D–SR time delays and rays of small launching angles, they are very sensitive to the SSP [22]. The striations according to n = 1, 2, 3 become steeper due to warm eddies and flatter due to cold eddies. In addition, according to the refraction law, a cold eddy reduces the speed of sound at the source, thus making the turning depths of first to third rays increase, while the opposite is true for a warm eddy.

If the source is fixed and the eddy moves, the sound speed perturbation would change the interference structure of the acoustic field. To investigate the changes in the acoustic interference structures during the eddy propagation, as shown in Figure 5, the CISs of the first three orders were extracted when cold and warm eddies propagated to different ranges from the source. Here the eddy is assumed to travel at a constant speed across the sound source; a Gaussian model simulated the mesoscale eddy, while other parameters remained unchanged. Figure 6a and 6b show the horizontal shifts of the constructive striations of the first three orders at a depth of 3300 m during the propagation of cold and warm eddies, respectively. As shown in Figure 6, the first striation exhibited the largest displacement (up to 1 km at this depth), while the second and third striations showed smaller shifts (0.6 km and 0.4 km, respectively). As the cold eddy approached the sound source from the periphery, the horizontal distance between the striations and the source progressively increased; conversely, this distance decreased as the eddy moved away. Except for the first few orders of interference striations, the positions of striations near the sound source remain almost unchanged. Figure 6b demonstrates the positional evolution of identical striations under warm eddy motion, revealing an inverse trend compared with the cold eddy. Distinct launch angle striations respond differently to eddy motion, fundamentally because ray paths at varying launching angles exhibit differential sensitivity to sound speed profile perturbations.

Traditional eddy identification primarily relies on satellite remote sensing to detect sea surface height anomalies (SSHAs) and closed temperature contours. While leveraging physics-based parameters and geometric contours for algorithmic eddy detection, these methods remain constrained by three key limitations: dependence on manual parameter tuning, the restrictive presumption of elliptical shapes and heightened sensitivity to SSH noise artifacts. Critically, they capture only sea surface information, lacking insights into subsurface eddy structure and dynamics. Acquiring structural data necessitates costly synchronous ocean surveys or sporadic Argo float deployments; neither approach enables sustained, large-scale, or repeated tracking and sampling of individual eddies. In contrast, the regular fluctuations in acoustic pressure energy reflect eddy movement and evolution, offering a novel pathway for eddy identification and tracking.

3.2. Spatial Frequency Domain

Usually broadband sources, such as ships of opportunity, are of interest for underwater acoustic detection and geoacoustic inversion. Thus, spatial frequency domain interference patterns are crucial for characterizing broadband acoustic fields. In this section, simulations are conducted to explore the impact of eddies on broadband interference striations, where source frequencies in the 50–300 Hz range and a receiver at 4950 m depth are used, and the other parameters are the same as those in Figure 2. As seen in Figure 7, the striations at each range before 30 km are periodic since the dominant striations of the receiver before this range are caused by interference between D and SR rays. The fine interference striations can be clearly seen after the range of 20 km, which are due to the interference between the D and the bottom refracted (BR) rays or that between the D and the surface bottom refracted (SBR) rays. Moreover, the periods of the interference patterns are the inverse of the multipath time delay. As shown in Figure 8, the multipath time delay decreases with the range, and then periods of interference striations increase with the range. By utilizing the inverse Fourier transform (IFT) of sound intensity, the multipath time delay differences can also be estimated from the peaks of the autocorrelation function of the received pressure signal, that is,

$$\begin{array}{c}\hfill \Gamma \left(\tau ,r\right)=F_f\left\{{\left|p\left(f,r\right)\right|}^2\right\}=E\left[p\left(t\right)p^{*}\left(x+\tau \right)\right]=R_{pp}\left(\tau ,r\right)\end{array}$$

(9)

where $R_{pp}\left(\tau ,r\right)$ represents the autocorrelation function of the pressure in the time domain. The result of $R_{pp}\left(\tau ,r\right)$ after performing the Fourier transform is shown in Figure 9. To enhance the time delay resolution, the source frequency range was expanded to [200, 500] Hz. Figure 9a displays the autocorrelation function $\Gamma$ for the receiver at 4950 m depth across various ranges under a cold eddy. Three smooth bright lines are observed in the autocorrelation function, corresponding to the time delay differences of D–SR, D–BR and D–SBR, respectively. In Figure 9b, the solid red line denotes the theoretical D–SR time delay for a warm eddy calculated using Bellhop, while the red scatter points represent secondary peaks extracted from the autocorrelation function (i.e., the warm eddy counterpart of Figure 9a). Similarly, the blue solid line and blue scatters denote the theoretical values and secondary peaks for the cold eddy. The close agreement between theoretical and extracted peaks is evident. The eddy-induced alterations in interference patterns arise because cold and warm eddies modify the sound speed profile, thus affecting the travel time of eigenrays and multipath delays. As the distance between the source and receiver changes, the temporal sequence of individual impulses typically remains unchanged, but their time intervals vary. Striation spacing in the frequency-range domain scales inversely with temporal intervals: a widening of the former induces a proportional contraction of the latter, and vice versa. Specifically, the cold eddy reduces sound speed, increasing time delay differences, while the warm eddy elevates sound speed, decreasing these differences. Such responses are encoded in the frequency-range domain interference striations, demonstrating the feasibility of inverting eddy parameters from these striations.

In the range frequency domain, the movement of an eddy also induces alterations in interference patterns. As shown in Figure 10a, striations of first to third are marked by black dashed lines. As the warm eddy core gradually propagates away from the source at distances of 0 km, 30 km and 60 km, a shift of interference striations toward lower frequencies is observed. During the process of the warm eddy leaving, the number of striations increases and the distance between striations decreases. The cold eddy generates the opposite effect. As shown in Figure 10b, striations of fourth to sixth move towards higher frequencies.

3.3. Arrival Structure

As demonstrated in Figure 5, Figure 6 and Figure 7 and Figure 10 above, the dynamic propagation of eddies induces measurable displacements in interference striations. Investigating changes in arrival structure at a fixed point during this process facilitates understanding its profound nature and underlying principles. When an eddy traverses a near-surface acoustic source, it modifies the local sound speed profile, thereby altering the refractive bending of sound rays. Consequently, a fixed receiver at the coordinates of (25 km, 4680 m) will experience dynamic variations in the received arrival structure (e.g., multipath components and time delays) as the eddy propagates. The experimental configuration illustrated in Figure 2a was designed with a mesoscale eddy initially approaching the source at constant velocity on the r-axis, subsequently receding. All the rays at this receiver are analyzed in this process, and the maximum launching and arriving angles of the rays and corresponding arrival angles and travel times are calculated by the ray model Bellhop. Then the dynamic behavior of interference striations under eddy-induced SSP perturbations is elucidated through analysis of temporal changes in arrival structure.

As illustrated in Figure 11c, with the sound source at the coordinate of (0 km, 100 m), the travel time of both D rays and SR rays gradually increases as a cold-core eddy approaches the source and decreases as it recedes. Similarly, the maximum launching and arrival angles and the time delays exhibit consistent trends during eddy advection. Owing to the bilateral symmetry of the Gaussian eddy model, the evolution of the arrival structure manifests pronounced symmetry. Notably, the curves of arrival structure exhibit slight asymmetries alongside this dominant symmetric pattern. This asymmetry occurs because the eddy induces stronger refractive bending of sound rays when its center lies between the source and hydrophone compared with when it is positioned on the opposite side of the source. Specifically, the cold eddy’s right side (proximal to the hydrophone) elevates the arrival angle slightly compared with its left side, while the warm eddy’s right side depresses the angle relative to its left counterpart.

During the migration of an eddy, two primary effects emerge: first, it alters the local sound speed at the source location; and second, it modifies the original refraction effects on outgoing rays, thereby changing the refraction experienced by acoustic rays during propagation. To isolate their individual contributions, a hypothetical range-independent scenario was modeled, where eddy-induced ray bending is suppressed, retaining only source-region SSP variations. As illustrated in Figure 12, the curve with refraction represents the integral effect of the Gaussian eddy, while the curve without refraction signifies only the local sound speed alteration induced by the eddy. The discrepancy between them quantifies the refraction effect. Due to the complete symmetry of the Gaussian eddy’s impact on the former, the strong symmetry observed in the arrival structure curve demonstrates that the former’s influence is dominant. This indicates that SSP perturbations near the source dominate arrival structure anomalies. When a cold eddy is positioned on either side of the acoustic source, it does not identically alter the refraction of acoustic rays. If the cold eddy center lies between the source and receiver, refraction effects slightly reduce the time delay and maximum launching angle. Conversely, when the cold eddy center is outside the source-receiver line, refraction increases these parameters. This insignificant change primarily stems from the small horizontal gradients of thermohaline anomalies induced by the eddy.

3.4. Robustness Analysis

Ocean systems are inherently complex composites of interacting multiscale dynamic processes. While mesoscale eddies exert modulation effects on acoustic fields, such effects in real ocean environments are inevitably perturbed by small-scale processes. Surface waves, driven by wind stress and ubiquitous across global oceans, introduce critical interference. Ignoring surface waves while assuming an idealized static sea surface fundamentally contradicts realistic ocean dynamic conditions. Concurrently, multiphysics robustness analysis constitutes a critical step in assessing engineering feasibility. To validate the universality of eddy-modulation models in authentic marine settings, surface wave disturbances are therefore introduced and discussed.

To investigate the impact of scattering effects caused by surface waves on interference striations, random sea surfaces were simulated based on the Pierson–Moskowitz (PM) spectrum. The operational workflow initiates with the initialization of wind speed, gravitational acceleration and spectral parameters, proceeding to calculate the peak frequency and define sea surface length alongside the minimum resolvable wavelength. Following this, the PM spectral model is constructed in the wavenumber domain. By incorporating random phases, a conjugate-symmetric spectral sequence is generated, synthesizing the initial sea surface elevation series through inverse Fourier transform. By varying wind speed to alter significant wave height and maximum elevation, different levels of sea surface roughness can be simulated.

Figure 13 demonstrates the disturbance effects of varying sea surface elevations on interference striations. Figure 13a shows the interference pattern under ideal specular reflection conditions. Figure 13b depicts the interference striation pattern when the maximum sea surface elevation change is 0.5 m, and the maximum sea surface elevation changes corresponding to Figure 13c,d are 1.6 m and 8.0 m, respectively. Due to surface wave scattering, the interference striations exhibit varying degrees of blurring and distortion. The blurring intensifies with increasing surface roughness. In contrast to Figure 13a, the interference fringes in Figure 13b remain virtually undisturbed. Compared with Figure 13a, the third to fifth interference striations in Figure 13c show some energy leakage. This is precisely the perturbation caused by scattering. Due to the unevenness of the sea surface, energy may be refracted into other regions. In Figure 13d, 7th to 13th striations demonstrate mutual overlapping, making it difficult to identify clear boundaries between them. Fortunately, an 8-m wave height represents an extreme and rarely occurring sea state. The severe deformation and overlapping observed in Figure 13d are thus rare occurrences. This demonstrates that perturbations at non-critical regions and smaller scales exert limited, even negligible, modulatory effects on interference striations.

4. Conclusions

The dynamic acoustic effects of mesoscale eddies represent a high-value yet challenging research topic. Understanding these effects is essential for optimizing sonar system performance through environmental matching and maximizing operational efficacy in marine environments. However, oceanographic survey methods used to track spatiotemporal variations in eddy position and intensity yield sparse data samples relative to the vastness of the ocean, limiting comprehensive characterization. Research on the modulation effects of mesoscale eddies on interference patterns is of significant importance for eddy parameter inversion, acoustic field propagation prediction and sonar performance optimization.

Regarding the impact of mesoscale eddies on acoustic propagation, previous studies have focused primarily on how these eddies affect acoustic transmission loss, yet they have not systematically investigated the detailed characteristics of the sound field. This paper demonstrates systematic variations in interference patterns within spatial and spatial frequency domains under eddy intrusion conditions, utilizing a Gaussian eddy model. When a warm eddy approaches the sound source, it causes the interference striations in the range frequency domain to shift toward higher frequencies. Conversely, when a cold eddy approaches the sound source, the interference striations shift toward lower frequencies. In the spatial domain, eddies mainly affect the striations at great depths at the edge of the DAZ. Simultaneously, a comparative analysis of displacement variations in distinct interference striations under eddy motion revealed that striations with smaller time delay differences exhibit larger displacements, while cold-core and warm-core eddies induce opposite-direction displacements. Additionally, the dynamic evolution of acoustic parameters during the migration of eddies was described. The results show that the changes in travel time, time delay and angle of arrival caused by cold and warm eddies are opposite. According to ray tracing theory, variations in time delay are consistent with alterations in interference striations, providing further mechanistic interpretation of striation variation phenomena. These conclusions establish a physical link between hydrological variations and acoustic field characteristics. Finally, the disturbance effects of surface waves on interference striations were analyzed to validate the robustness of the model and conclusions in complex environments.

While this work advances our understanding of eddy–acoustics interactions, three key limitations warrant attention. First, the scenario where strong warm eddies induce dual sound channeling remains unexplored. Such configurations could fundamentally alter interference striation patterns through modal coupling mechanisms [46]. Second, our analysis does not quantify mesoscale eddy impacts through waveguide invariant distributions—a formalism that could provide spectral domain metrics for striation slope stability under eddy perturbation. Finally, these findings are based on idealized eddy models and may not fully capture real-world complexity, which encompasses eddy asymmetry, nonlinear dynamics and temporal variability.