FTLE serves as a quantifier for the maximal rate of separation between trajectories over a finite time period, effectively interpreting LCS within various flow fields. The computation of FTLE begins by tracking the movements of tracer particles within a fluid, integrating their positions over a selected time interval. This process involves mapping the trajectory of a set of uniformly distributed tracer particles, which are initially located at position X at time t ₀, to their positions after a duration T (Beron-Vera et al., 2008; Harrison and Glatzmaier, 2012). The mapping, described by the transformation φ t 0 t 0 + T ( X ) , plays a critical role in understanding the dynamic behavior and shape changes of fluid parcels. To quantify the time-dependent deformation of fluid parcels over time, the Cauchy-Green deformation tensor, represented as Δ , is employed. This tensor is computed using a finite-difference approximation of the derivative of the transformation φ t 0 t 0 + T ( X ) with respect to position X . This approximation is essential for estimating derivatives at discrete points and involves comparing the positions of neighboring particles at times t 0 and t 0 + T . For 3D flows, this comparison includes the positions of six neighboring particles (see Figure 1 ) to approximate the sensitivity of the particle’s trajectory to its initial conditions. The formation of this tensor is represented in Equation 1, and the finite-difference approximation is shown in Equation 2.

where ( x i , j , k ,   y i , j , k ,   z i , j , k ) denote the positions of particles in a 3D grid, with indices i , j , k corresponding to their grid positions. For 2D flows, only the first 2 × 2 minor matrix of the tensor is relevant, considering the four nearest particles (Haller, 2015).

The FTLE, denoted as σ t 0 T   ( q ) , is then derived from the maximal eigenvalue, λ m a x   ( Δ ) , of this deformation tensor. It is calculated using the equation (Haller, 2001a):

The parameter of T can assume both positive and negative values. Specifically, positive T values correspond to forward temporal separations, leading to the identification of repelling LCS. Conversely, negative T values are indicative of backward temporal separations, resulting in the detection of attracting LCS in a forward temporal framework (Haller, 2001a, 2002; Shadden et al., 2005). It is worth noting that the choice of interval time T , to accurately capturing meaningful flow structures and dynamics, depends on the characteristics of the fluid domain being studied and the specific objectives of the analysis. Primarily, the predominant time scales inherent to oceanic dynamics must be recognized, which often manifest as tidal, diurnal, or even seasonal variations (Huhn et al., 2012; Bakhoday-Paskyabi, 2020). The chosen FTLE interval ought to be matching with these time scales to ensure an accurate representation of oceanic motions. The overall research objective further defines the choices. Long-term FTLE evolutions require longer intervals (Bakhoday-Paskyabi, 2020; Paul and Sukhatme, 2020; Rypina et al., 2020; Dargahi, 2022), while transient phenomena require shorter spans (Castelle and Coco, 2013; Lee et al., 2013). Evaluating particle trajectory separations across various regions provides empirical insights into time interval selection (Shadden et al., 2009; Huhn et al., 2012). Meanwhile, analysis of the probability distribution function (PDF) of the FTLE field can highlight the dominant dynamics. Distinct peaks in the PDF suggest consistent behavior across the marine domain and indicate a suitable interval choice (Huhn et al., 2012). Sensitivity analyses, which involve calculating FTLE fields over various intervals, are useful in determining the point at which LCS remains consistent (Huhn et al., 2012).

Upon computation, high FTLE values (FTLE ridges) serve as indicators, highlighting areas where particle trajectory separations are particularly pronounced. This distinction in the FTLE field provides a robust mechanism to detect regions of significant dynamical behavior, emphasizing areas of the flow domain where material lines experience accelerated stretching or filamentation (Allshouse and Peacock, 2015b). It’s also worth noting that areas distanced from FTLE ridges tend to exhibit minimal activity (Zhong et al., 2022).

A comprehensive analysis of the FTLE field requires both forward-time and backward-time FTLE computations. However, when analyzing both stretching and filamentation material lines, the overlapping of forward and backward FTLE fields can lead to interpretative challenges. Addressing this, d’Ovidio et al. (2004) integrated normalized forward and backward FTLE fields, thereby introducing the concept of hyperbolic FTLE fields as shown in Equation 4:

where FTLE max + is the maximum of the forward FTLE and the FTLE max − is the maximum of the backward FTLE. This innovative approach offers a finite-time generalization of the traditionally recognized concept of normally hyperbolic invariant manifolds in dynamic systems, providing clarity in the visualization and understanding of complex flow behaviors. Figure 2 interprets these concepts using a hyperbolic LCS representation. In Figure 2 , the inflowing manifolds, shown by the red line, signify repelling LCS, while the outflowing manifolds, illustrated by the blue line, represent attracting LCS. The intersection of these two lines is the saddle point. Flow direction proximity to this hyperbolic pattern is indicated by the blank arrows in Figure 2 . Inferring flow direction from LCS is important. As illustrated in Figure 2A , trajectories associated with the red line are drawn towards the saddle point, while those along with the blue line move away from it. Consequently, as the flow approaches the saddle point via the red line and subsequently departs along the blue line, the change of flow direction is observed. In the nearshore area, as shown in Figure 2B , when the hyperbolic LCS intersects with the shoreline, the shoreline assumes the role of a saddle point. Flow approaches the shoreline along the red line and away from the shoreline via the blue line (Huhn et al., 2012).

The dynamics of stretching and filamentation in ocean fluids are inherently associated with the presence of LCS lines (Peacock and Dabiri, 2010; Peacock and Haller, 2013). Specifically, attracting LCS regions tend to draw fluid trajectories closer together, which can align and sometimes compress nearby material lines, making them more compact in the direction perpendicular to the stretching. Conversely, repelling LCS regions push trajectories outward, leading to the elongation and thinning of material lines as they are pulled apart (van Sebille et al., 2018). Consequently, these LCS lines act as transport barriers, inhibiting any flux across them (Shadden et al., 2005; Bettencourt et al., 2012; Huhn et al., 2012; Zhong et al., 2022). Wei et al. (2018) introduced two distinct particle sets on either side of LCS and monitored their paths over a span of 144 hours, it was documented that the majority of the simulated particles remained confined to their initial sides of LCS.

To better visualize and understand the behaviors of complex flow, various methods have been employed to extract LCS lines from FTLE fields (Garth et al., 2007; Sadlo and Peikert, 2007; Sadlo et al., 2011; Lipinski, 2012). A widespread approach involves setting a threshold to highlight regions where FTLE values exceed the specific value (Lipinski, 2012; Suara et al., 2020; Zhong et al., 2022). While the selection of an optimal threshold remains non-definitive, empirical investigations indicate that thresholds between 50 – 80% of the maximal FTLE value offer a reliable representation (Lipinski, 2012). Several research indicated that higher threshold values might narrow down the areas of significant activity without drastically altering the overall flow pattern (Rockwood et al., 2019; Giudici et al., 2021). The threshold selection is inherently subjective and should align with the analysis goals, the unique characteristics of the flow, and the level of detail and robustness required for identifying coherent structures.

Shear and confluence are fundamental components in interpreting complex dynamics and motions. Shear is characterized by velocity differentials between neighboring oceanic layers. Confluence, in contrast, refers to the stretching of water masses with differing attributes (Biron et al., 1993). Backward FTLE ridges (attracting LCS) were utilized to visualize shear and confluence regions (Haller, 2002; Gough et al., 2016). Gough et al. (2016) posited that areas with high FTLE values correspond with pronounced gradients in the velocity field, signaling regions of shear. Similarly, zones where LCS accumulate or are densely packed suggest areas of particle confluence. Through the temporal and spatial analysis of LCS, researchers are able to deduce the existence and dynamics of shear zones and confluence regions on the ocean’s surface (Gough et al., 2016).

Both shear and confluence can enhance or create strong horizontal gradients, leading to the formation of fronts (Owen, 1981; Gough et al., 2016). In oceanographic studies, fronts stand as a keystone of marine hydrodynamics, serving as the boundaries between different water masses (Mathur et al., 2019). These fronts play an important role in shaping the dynamics within marine systems and significantly influence global climatic interactions (Gough et al., 2016; Liu et al., 2018; Mathur et al., 2019). As an advanced analytical tool, FTLE and LCS were employed by Mathur et al. (2019) to explain the orientation of thermal fronts in the north Bay of Bengal from December 2015 to March 2016, as shown in Figure 3 . These thermal fronts, presenting the boundary between seawater and freshwater from river runoff, were identified by locating regions where the magnitude of the spatial temperature gradient exceeds a specified threshold of 0.02°C/km (Mathur et al., 2019). The sea surface temperature (SST) field illustrated a prominent thermal front extending from around 14°N to 19°N at around 86°E, which was labelled as C-front ( Figure 3B ). This specific C-front feature exhibited similarity with the backward FTLE, as shown in Figure 3C . Furthermore, overlaying the extracted attracting LCS onto the SST field ( Figure 3D ) revealed a robust association between the location/orientation of thermal fronts and the attracting LCS in ocean surface currents. For instance, a concurrence between the attracting LCS and extensive C-front feature (about 700 km) was observed. Subsequent analyses revealed the persistence of this similarity between the attracting LCS and the thermal front over an extended duration of at least one month. Moreover, Mathur et al. (2019) introduced the angle α ∈ [ 0 , π 2 ] for quantification purposes. The trajectory of thermal fronts is hypothesized to be orthogonal to the vector of ∇(SST), while the direction of the attracting LCS corresponds to the eigenvector associated with the maximum eigenvalue of the Cauchy-Green strain tensor. The outcomes, as shown in Supplementary Figure 1 , highlighted that alignment between thermal fronts and proximal attracting LCS was particularly prominent in large-scale frontal features and any frontal feature if it located near a sufficiently robust attracting LCS. However, certain limitations, such as the reliance on 2D modeling data, potential errors in SST-gradient-corrections, and possible inaccuracies in satellite-based observations, should be acknowledged (Mathur et al., 2019).

Fronts can appear individually, in clusters, or conjunction with eddies throughout the oceans, especially within the surface layer (Owen, 1981). Eddies are localized circular current systems, which can be either cyclonic or anticyclonic (Chelton et al., 2011). Characterized by their vortical motion, these eddies play a pivotal role in trapping and transporting heat, nutrients, and contaminants within the marine environment (McWilliams, 2008). The application of FTLE-based LCS technique has become increasingly prevalent for examining the formation and dissipation of eddy, and the consequent implications on fluid mixing (Waugh et al., 2006; Waugh and Abraham, 2008; Prants et al., 2011, 2015; Gough et al., 2016; Liu et al., 2018; Paul and Sukhatme, 2020; Rypina et al., 2020; Zheng et al., 2020; Filippi et al., 2021). For example, Zheng et al. (2020) defined typical eddy activities and flow dynamics in the vicinity of the Luzon Strait using LCS framework. Employing a backward FTLE approach, LCS was extracted and subsequently compared against satellite-tracked drifter trajectories from corresponding dates. Notably, the observed LCS consistency with drifter patterns emphasizes its ability to define transport trajectories and determine boundaries within the investigated domain. Examination of FTLE fields revealed four major transport patterns proximal to the Luzon Strait, as shown in Figure 4 . The first pattern presents a typical Kuroshio “leaking” trajectory, with LCS extending northwestward from the eastern precincts of Luzon Island into the South China Sea (SCS) ( Figure 4A ). The second pattern represents the northward “leaping” pattern of Kuroshio, the corresponding FTLE fields in Figure 4B display LCS along the Kuroshio trajectory, extending more distinctly northward compared to those shown in Figure 4A . The type of “looping” LCS near the Luzon Strait is clearly seen in the FTLE field in Figure 4C , which makes the water loop or excurse into the SCS and then turn clockwise back to the Pacific. The fourth type of pattern, called “outflowing” LCS, demonstrates the outflow of SCS waters into the Pacific, with the associated LCS emphasizing FTLE ridges extending northeastward from the western regions of Luzon Island to the eastern of Taiwan Island ( Figure 4D ). Additionally, these observed transport patterns show seasonal variations. Among them, the “leaping” pattern, predominant across seasons, peaked in spring, while “leaking” and “outflowing” patterns exhibited sensitivity to the monsoon reversals, prevalent in winter and summer, respectively. Interestingly, the “looping” pattern remained consistent across all seasons. On an annual scale, the leaping pattern’s frequency dominated, with notable deviations in 1998 and 2016 (Zheng et al., 2020).

In addition to surface eddies, the vertical eddies structure also plays an important role in enhancing the precision of simulations and predictions in oceanographic modeling. Prants et al. (2015) delved into the vertical structure of mesoscale anticyclonic eddies in the Japan/East Sea using an ocean circulation quasi-isopycnal layered model. By employing the FTLE fields, they highlighted the nonlinear stability of these eddies and their ability to reveal the complex pathways and barriers that determine transport and mixing processes at mesoscales and submesoscales. Their examination extended up to 250 meters deep, segmented into ten layers, with the 1^st, 3^rd, 5^th, and 9^th layers highlighted in Figure 5 . A key observation was the presence of elliptic stagnation points at the eddy centers, characterized by zero velocity, particularly discernible in layers below the fourth. Through the FTLE field, the study detailed the eddy’s horizontal layer-by-layer structure, with “ridges” signifying areas of maximal particle filamentation. Such insights are crucial, suggesting that while particles on one side partake in vortex activities, those on the opposite move away from the eddy. Prants et al. (2015) also established that the simulated anticyclonic eddy underwent a seasonal transition. The structure at the subsurface during summer reached the surface by fall. This transitional behavior can be attributed to the variances in seasonal stratification, especially evident in the pycnocline. Moreover, their findings’ accuracy is validated through comparisons with a 1999 oceanographic survey (Talley et al., 2001, 2006), emphasizing its alignment with real-world observations (Prants et al., 2015).

To clarify the intensity and spatial distribution of turbulent eddy phenomena, the metric Eddy Kinetic Energy (EKE) was introduced to quantify the kinetic energy inherent to mesoscale eddies within fluidic systems, as given by the Equation 5 (Waugh et al., 2006):

where   u ′ and v ′ represent temporal anomalies in the zonal and meridional velocities, respectively (i.e., deviations from their temporal mean), while angle brackets signify temporal averaging. Research has validated that FTLE effectively describes characteristics inherent to EKE on both localized (Waugh et al., 2006; Paul and Sukhatme, 2020) and global scales (Waugh and Abraham, 2008). For instance, Paul and Sukhatme (2020) created the FTLE maps across varying seasons, including February-March-April (FMA), June-July-August-September (JJAS), and October-November-December (OND). The patterns within these FTLE maps paralleled those of EKE for each season, with elevated EKE correlating with zones of strong stirring, underlining the efficacy of FTLE in revealing stirring intensities (Waugh et al., 2006; Waugh and Abraham, 2008; Paul and Sukhatme, 2020). Additionally, Waugh et al. (2006) introduced a strain rate to describe the deformation rate of fluid elements in flow, along with EKE for comparative analysis with FTLE fields. The strain rate is represented as shown in Equation 6 (Waugh et al., 2006):

where ( u ,   v ) represents the total velocity field. The relationship between mean FTLE, strain rate, and EKE has been systematically quantified, as shown in Figure 6 . In this depiction, individual markers denote individual grid points, while the boxes capture the mean values over 5° by 5° subregions. Notably, as indicated in Figures 6A, B , in regions characterized by higher strain rates, the corresponding FTLE values are consistently lower. The decrease in FTLE values in regions with higher strain rates is due to particles temporarily residing in areas of high strain. As the integration time increases, these particles move to areas with smaller strain rates. Consequently, the overall stretching rate (FTLE values) diminishes when compared to the initially higher strain rate (Waugh et al., 2006). On a global scale, Waugh and Abraham (2008) presented a comparative analysis of FTLE, strain rate, and EKE ( Supplementary Figure 2 in supplemental materials). This comparison emphasizes the inherent relationships among these metrics, with their spatial distributions showcasing considerable overlap (Waugh and Abraham, 2008). The consistent representation of FTLE across various temporal scales, supported by EKE and strain rate analysis, highlights the importance of these metrics in interpreting marine dynamics.

The FTLE has proven to be a critical tool in ocean fluid dynamics, particularly for revealing complex phenomena such as shear, confluence, oceanic front, and eddy. While traditionally used in a standalone capacity, FTLE’s effectiveness is further amplified when used in conjunction with metrics like EKE and strain rates in comparative analyses. This combined approach allows for a more comprehensive understanding of ocean fluid dynamics, offering a robust framework for exploring and interpreting the intricate dynamics of oceanic flows, thus significantly contributing to the advancement of marine research and modeling.

Tidal forces and the resultant currents are influential in the generation of turbulence and mixing. These dynamics profoundly influence the transport of both dissolved and particulate substances, inducing periodic oscillations in the physical and chemical properties (Mao et al., 2004). Given the impact of tidal forces on the movement and patterns of water flow in the oceanic environment, it becomes vital to delve deeply into the influence of tides in these contexts (Wei et al., 2018; Suara et al., 2020; Filippi et al., 2021; Zhong et al., 2022). Wei et al. (2018) compared LCS during varied tidal phases within the Pearl River Estuary (PRE). Their comprehensive mapping, as shown in Figure 7 , represented the hourly evolution of LCS over a tidal cycle. The evolution of LCS was periodic, aligning with the alternating ebb and flood tidal currents ( Figures 7A-O ). This periodicity approximated a duration of 12.5 hours, which is consistent with the predominant tidal phase in the PRE. During ebb tide, an obvious repelling LCS was formed in the high dynamic area and separated the mid-estuary into two parts, as highlighted by a blue quadrangle in Figure 7B . This LCS line formed after high tide, persisting for 6 to 7 hours, dissipating just prior to low tide ( Figures 7J-N ) (Wei et al., 2018). Similarly, Filippi et al. (2021) employed FTLE to investigate transport mechanisms during different tidal phases within Scott Reef. They recommended that the repelling/attracting LCS resulted from the filamentation/stretching. Their findings illustrated that tidal forces had a significant influence on LCS formation within the studied region. Specifically, the stretching was strongest during the high tide phase at 6:00 UTC. This stretching then undergoes a systematic decrease throughout the ebb period. Furthermore, they observed minimal filamentation/stretching at the low tide at 12:00 UTC, followed by an increase in filamentation during the flood period.

Wind-driven processes significantly influence oceanic surface transport through the additional drag force they exert, which propels the movement of floating materials directly at the ocean’s surface. FTLE and LCS have been at the forefront of a new perspective on understanding the wind’s role in oceanic material transport (Huhn et al., 2012; Gough et al., 2016; Allshouse et al., 2017; Wei et al., 2018; Suara et al., 2020). For instance, Huhn et al. (2012) investigated the influence of wind on ocean surface transport in Ria de Vigo by extracting the repelling and attracting LCS under southern and northern wind conditions, respectively, as shown in Figure 8 . Under southern wind conditions ( Figure 8A ), the repelling LCS separates the flow at the Cies Islands (L1) and Cape “Cabo Home” (L2), facilitating the transport of water in-between L1 and coast into the Ria de Vigo at the south, and other water bodies leave the estuary through the north mouth. Moreover, the attracting LCS (L3) connected to the north of Cies Island suggests that this outflow remains coastal, progressing northward into the Ria de Pontevedra. Conversely, during northern wind conditions ( Figure 8B ), the flow largely inverts. An attracting LCS (L4) restricts surface water entry into the inner estuary from the north. Although there exists some variability in LCS during different wind directions, they consistently represent model flow patterns under similar meteorological conditions. Consequently, wind is recognized as an indispensable parameter in utilizing LCS methodology to assess marine water transport dynamics. Such a premise is corroborated by the findings of Allshouse et al. (2017) and Suara et al. (2020).

Gough et al. (2016) also investigated the influence of wind forcing on the formation of LCS by comparing LCS identified through High-Frequency (HF) radar-derived velocities and patterns observed in satellite SST imagery. They found that during periods of light winds, LCS appeared to be influenced by circulation beneath the surface mixed layer. However, in conditions of moderate upwelling winds, there were noticeable shifts between LCS and SST fields, suggesting a decoupling of the surface mixed layer from the underlying water circulation. This decoupling became even more pronounced during strong winds, leading to a complete separation of the surface mixed layer from the circulation below, indicating that the surface currents captured by HF radar predominantly aligned with the wind-induced dynamics rather than the patterns in the SST imagery. Such findings underscore the complexity of interactions between surface and subsurface oceanic layers, and the significant role of wind conditions in these dynamics. The study highlights the value of LCS in understanding water movement patterns, providing insights that are not readily obtainable from SST images alone (Gough et al., 2016).

Tidal forces and wind-driven processes significantly shape ocean fluid dynamics, influencing the mixing and the transport of substances. Ser-Giacomi et al. (2015) introduced a novel method to study geophysical fluid transport, integrating the FTLE method with network theory. This approach establishes a direct correlation between the node degree in a flow network and the local fluid stretching, traditionally quantified by FTLE. The method uniquely applies FTLE to inform flow network construction, linking the nodes’ in-degree and out-degree to backward and forward FTLE values, respectively. This reflects the dispersion and mixing potentials of fluid parcels. The combination of FTLE and network theory offers a comprehensive perspective on fluid transport, merging local dynamic quantification with a structural overview of system-wide connectivity and pathways. This innovative fusion enhances the understanding of fluid motion, identifying key dispersion regions and major mixing zones in the fluid transport system (Ser-Giacomi et al., 2015).

Tidal forces and wind-driven processes are integral in shaping ocean fluid dynamics, notably affecting substance mixing and transport. The impact of tidal forces on water flow patterns is marked by periodic oscillations. Understanding these tidal influences is vital, as demonstrated through the use of FTLE to visualize changes in tidal phases and their corresponding effects on water movement. Similarly, wind plays a pivotal role in oceanic surface transport, altering the movement of floating materials through additional drag forces. The application of FTLE provides insights into how wind affects oceanic material transport. These approaches have revealed the complex interactions between surface and subsurface oceanic layers under varying wind conditions, highlighting the critical role of wind in influencing marine dynamics. Overall, the understanding of both tidal and wind forces, facilitated by FTLE-based LCS, is essential in comprehensively interpreting ocean fluid dynamics.

The applications of FTLE-based LCS analysis in coastal zones have provided profound insights (Lekien et al., 2005; Bakhoday-Paskyabi, 2020; Suara et al., 2020). For example, Bakhoday-Paskyabi (2020) employed FTLE-based LCS method to evaluate barriers impeding the spread of ocean contaminants and the distribution of marine populations in the Lofoten area of the Norwegian Sea. Their observations revealed that LCS lines within the region between 67° – 70° N and 10° – 15° E acted as a barrier to the dispersion of pollutants, dynamically separating the coastal and open-ocean water masses (Bakhoday-Paskyabi, 2020). Further delving into the seasonal variability of LCS ( Figure 9 ), they suggested that the seasonal variability of surface currents, freshwater influx from riverine sources and precipitation, and water depth, are the key factors for LCS formation (Bakhoday-Paskyabi, 2020). Beyond the 70°N latitude, transport barriers during the winter months appeared to direct pollutants toward the Barents Sea (Bakhoday-Paskyabi, 2020). However, this effect is reduced during summer, and shelf waters can easily move into deeper Arctic waters (Bakhoday-Paskyabi, 2020). The seasonal variation of LCS was also studied by Giudici et al. (2021). They identified a strong correlation of LCS with specific wind patterns in different seasons, which informs the prediction of pollutant trajectory (Giudici et al., 2021).

The industrial infrastructures are commonly located in coastal regions, and the effluents discharged from factories represent a primary source of marine pollution. Therefore, Lekien et al. (2005) employed LCS to create a pollutant release strategy aimed at minimizing the impact of industrial effluents on the coastal environment. They utilized the VHF-radar-derived surface current data from the Florida coast to compute the forward FTLE fields, which identified a sequence of repelling LCS lines. These LCS originated from the coast and extended in a southward direction, as illustrated in Figure 10 . LCS acts as a material barrier separating the southwest and northeast regions. Therefore, particles originated in the northeast of this barrier (white parcel) are rapidly spread within hours. In contrast, particles originating southeast of the barrier (black parcel) tend to undergo multiple re-circulation events near the Florida coast prior to their eventual integration with the currents. Notably, the position of LCS exhibits temporal variations. As such, industrial and sewage facilities should not release effluents when LCS is situated to the north of their discharge pipelines. According to the temporal variation of LCS, researchers proposed an optimal effluent release timetable, which can diminish the environmental impact of pollutants in the coastal zone by a factor of three, compared to a consistent temporal release (Lekien et al., 2005).

Marine debris contamination is emerging as a critical issue in the area of marine pollution. A thorough understanding and predictive ability regarding the sources, pathways, and exchange mechanisms of this debris are essential to address the issue effectively. Studies have identified FTLE-based LCS as a powerful tools to handle these challenges (Kim et al., 2012; Ourmieres et al., 2018; Suara et al., 2020; Ghosh et al., 2021). As a case in point, Suara et al. (2020) utilized the FTLE method to predict the fate of marine debris in the tidal embayment of Moreton Bay (MB). The extraction of LCS from the FTLE was achieved using a threshold, such that only FTLE values exceeding 50% of the maximum value were considered as LCS. Subsequent findings revealed that most LCS are attached to islands within MB, as illustrated in Figure 11 . These LCS findings are consistent with the documented debris source data (Suara et al., 2020). Weak LCS lines were observed in Deception Bay (DB), coincidently quite much marine debris (32%) was observed in DB. However, only 16% and 7% of marine debris were detected in Redcliffe Headland (RCH) and Bramble Bay (BB) where had strong LCS lines. These suggested that the transport of marine debris cannot cross LCS lines (Suara et al., 2020). They also suggested that the presence of islands and headlands within the embayment critically influences LCS structure, subsequently affecting the transport and dispersion of debris. This observation was consistent with subsequent studies where LCS was employed to assess the persistency of debris accumulation in MB (Ghosh et al., 2021). By measuring LCS during the cumulative duration of time (14 days), most detected debris accumulation zones were identified as proximal to the islands and headlands of MB (Ghosh et al., 2021). Furthermore, both Suara et al. (2020) and Ghosh et al. (2021) evaluated the impact of tidal phases and wind dynamics on LCS formations in MB, noting the variability in structure and saddle point locations across different tidal phases. Their observations also highlighted wind as a factor amplifying the contraction and expansion rates near barriers, decreasing mixing intensities, and altering particle transport direction (Suara et al., 2020; Ghosh et al., 2021).

The increase in petroleum product consumption and transportation has raised concerns about potential oil spills. Consequently, recent studies have employed the FTLE and LCS methods to predict oil spill trajectories (Ivić et al., 2017; Gough et al., 2019; Zhong et al., 2022). For instance, Zhong et al. (2022) utilized forward and backward FTLE fields to evaluate the influence of tidal dispersion on oil spill trajectory. By normalizing and combining the backward and forward FTLE fields with Equation 3 in Section 2.2, they generated the hyperbolic FTLE field. This field was compared with the real situation of the M/V Marathassa oil spill on April 8, 2015 (Butle, 2015). Given the spill started at 14:00 during a mid-ebb phase of the spring tide (Zhong et al., 2018), the corresponding FTLE field was compared with a recorded situational map from the City of Vancouver (Stormont, 2015). This comparative analysis, illustrated in Figure 12 , robustly represented the actual dispersion patterns of the M/V Marathassa oil spill. For instance, Figure 12A demonstrates repelling LCS lines (in red) linking the Outer Harbour’s center to its eastern end, correlating with observed oil deposits at English Bay Beach and Siwash Rock ( Figure 12B ). Additionally, the migration of oil from the Outer Harbour’s north coast to the Inner Harbour via First Narrows during April 9 – 10, 2015, is corroborated by attracting LCS lines (in blue). Notably, no oil was observed on the southern shoreline of the Outer Harbour, aligning with the east-west orientation of repelling LCS lines preventing southern transportation. Essentially, these FTLE insights afford a comprehensive understanding of the M/V Marathassa spill dynamics. Zhong et al. (2022) also integrated a stochastic approach via the Oil Spill Contingency and Response (OSCAR) model to determine contamination probabilities at six potential spill locations in Burrard Inlet. This model’s outcomes fit with the FTLE-derived interpretations, underscoring FTLE’s effectiveness as a powerful tool for predicting oil spill trajectories and supporting spill response strategies (Zhong et al., 2022).

Overall, the application of FTLE-based LCS analysis in coastal regions has significantly advanced our understanding of pollutant dispersion. Researchers have been able to identify barriers that block the spread of contaminants and influence the distribution of pollutants. Seasonal variability in LCS, influenced by factors such as surface currents, freshwater influx, and water depth, further affects pollutant dispersion patterns. In addition, the role of wind and tidal forces in shaping LCS structures highlights their impact on debris transport and accumulation in coastal embayment. These insights are crucial for understanding pollutant trajectories and developing effective strategies for environmental protection. The integration of LCS with other models, such as oil spill contingency and response simulations, demonstrates the potential of these tools in predicting and managing marine pollution incidents, making them indispensable in marine environmental research.

Sea ice serves as a critical regulator in the energy exchange among the atmosphere, cryosphere, and ocean. Documented decreases in sea ice coverage, coupled with a rise in younger/thinner ice, underscore a concerning trend (Rothrock et al., 1999; Rignot et al., 2008; Landy et al., 2022). The younger/thinner ice is more sensitive to processes of deformation and breakage, which could subsequently change the atmospheric circulation patterns (Moore et al., 2018) and midlatitude weather (Siew et al., 2020). This highlights the urgency of precise monitoring of sea ice dynamics to comprehend the nuances of sea ice deformation, fracturing, and refreezing, as well as to assess their broader implications for global climate and maritime infrastructure.

Several studies have employed FTLE methods to investigate the dynamic of sea ice (Szanyi et al., 2016; Kumar et al., 2018). For instance, Szanyi et al. (2016) utilized the FTLE field to explore the dynamical shifts in the Arctic sea ice from November to April during 2006 to 2014 (Szanyi et al., 2016). Their analysis showcases three prominent regions of strong deformation, evident through clear LCS lines, which were consistent with the multi-year ice coverage data. Moreover, they introduced the Lagrangian Averaged Divergence (LAD) to distinguish the relative contributions of filamentation, shear and hyperbolic structures. The filamentation-induced ice deformation was observed in the Beaufort Sea, while shear predominantly influenced the central Arctic. FTLE ridges in the central Arctic exhibited influence from both shear and filamentation (Szanyi et al., 2016). Further comparison of FTLE results with multi-year ice (MYI) presented a noticeable correlation between high FTLE values and decreased MYI coverage (Szanyi et al., 2016).