Lagrangian Coherent Structure Research Articles

On the evidence of helico-spiralling recirculation within coherent cores of eddies using Lagrangian approach

Oceanic eddies exhibit remarkable coherence and longevity compared to other transient features in the surrounding flow. They possess the ability to transport properties over extensive distances while maintaining their material identity intact. The Lagrangian Coherent Structure (LCS) framework has proven effective in capturing these coherent eddies, where they display a solid-body-like rotation. Although various LCS approaches have been employed to investigate different facets of coherent eddies, a comprehensive understanding of their three-dimensional structures and internal dynamics remains elusive. This study aims to advance our comprehension of coherent eddies’ structural characteristics and delve into the precise nature of their internal dynamics by utilizing the Lagrangian Averaged Vorticity Deviation approach. Two eddies, one cyclonic and the other anti-cyclonic, were chosen from a high-resolution simulation carried out in the Bay of Bengal using the Regional Ocean Modeling System (ROMS). The findings unveil that these eddies have three-dimensional coherent cores resembling gently tapered cones that are broader at the surface and gradually narrow towards the bottom. Intriguingly, the dynamically coherent core of these eddies exhibits simultaneous upwelling and downwelling while maintaining their volumes during advection due to persistent material coherence. The three-dimensional trajectories followed by the fluid parcels inside the coherent core are helical. Their two-dimensional horizontal projections show alternating spiral bands of upwelling and downwelling which are the manifestations of Vortex Rossby Waves. These observations lead to a conceptual framework of a three-dimensional helico-spiralling recirculation pattern within the coherent cores of eddies.

Quantifying the role of submesoscale Lagrangian transport features in the concentration of phytoplankton in a coastal system

Abstract Food resources in the ocean are often found in low densities, and need to be concentrated for efficient consumption. This is done in part by oceanographic features transporting and locally concentrating plankton, creating a highly patchy resource. Lagrangian approaches applied to ocean dynamics can identify these transport features, linking Lagrangian transport and spatial ecology. However, little is known about how Lagrangian approaches perform in ageostrophic coastal flows. This study evaluates two Lagrangian Coherent Structure metrics against the distribution of phytoplankton; Finite Time Lyapunov Exponents (FTLE) and Relative Particle Density (RPD). FTLE and RPD are applied to High Frequency Radar (HFR) observed surface currents within a biological hotspot, Palmer Deep Canyon Antarctica. FTLE and RPD identify different transport patterns, with RPD mapping single particle trajectories and FTLE tracking relative motion of paired particles. Simultaneous measurements of circulation and phytoplankton were gathered through the integration of vessel and autonomous glider surveys within the HFR footprint. Results show FTLE better defined phytoplankton patches compared to RPD, with the strongest associations occurring in stratified conditions, suggesting that phytoplankton congregate along FTLE ridges in coastal flows. This quantified relationship between circulation and phytoplankton patches emphasizes the role of transport in the maintenance of coastal food webs.

A Lagrangian coherent eddy atlas for biogeochemical applications in the North Pacific Subtropical Gyre

Abstract. Mesoscale eddies affect phytoplankton in several ways, including the horizontal dispersal and mixing of populations. Coherent eddies trap and contain fluid masses, whereas other eddies mix more freely with surrounding waters. To evaluate the role of lateral dispersal and trapping on the biogeochemical properties of eddies, we must accurately characterize their coherency. We employed a Lagrangian approach to identify materially coherent structures in remote sensing observations and developed a methodology to track them over their entire individual lifetimes. We provide an atlas of two decades of coherent eddies with an 8 d resolution in an intensely studied region of the North Pacific Subtropical Gyre (Jones-Kellett, 2023a, https://doi.org/10.5281/zenodo.8139149). The atlas was specifically designed to facilitate biogeochemical applications and was synchronized with available ocean color products. We identified coherent features using backward Lagrangian trajectories because the recent history of coherency (rather than the future coherency) will be most valuable for interpreting associated biogeochemical signatures. We compared the atlas of Lagrangian coherent eddies with an atlas of Eulerian eddies identified using the more conventionally used Sea Level Anomaly method. Although 65 % of tracked Sea Level Anomaly eddies are classified as coherent at some point in their lifetime, only 54 % contain a Lagrangian coherent structure at any given time. We found similar variations in the temporal and spatial distributions of coherent and Sea Level Anomaly eddies. However, strictly coherent eddies have a clearer relationship between size and longevity and form distinct regional regimes based on polarity. We illustrate the potential of the coherent eddy atlas for biogeochemical applications by examining the relationship between bloom development and eddy evolution in a case study of a Hawaiian Lee cyclone.

On the residence enhancement mechanism of the cavity-based dual-mode scramjet combustor through the Eulerian and Lagrangian analysis

The lower total temperature and pressure in a dual-mode scramjet engine lead to slower rates of evaporation and chemical reactions, while the inflow velocity is higher than that in a ramjet engine. Consequently, enhancing fuel residence time becomes a more critical challenge. The cavity is a crucial device for enhancing the residence time. However, the quantitative residence capacity and the mechanism have not yet been revealed, especially in wide-range speed inflow conditions. This work employs the delayed detached eddy simulation method to investigate the mass transport and fluid residence characteristics of the wide-range subsonic flow (Ma = 0.3, 0.4, 0.5, 0.6, 0.7) over the cavity. The Lagrangian coherent structure is utilized to characterize the dynamic evolution of the large-scale vortex in the cavity shear layer. Particle tracking is employed to delicately determine the net mass exchange rate and the quantitative cavity residence time. Based on the entrainment process of the large-scale vortex and mass exchange between the mainstream and cavity, this paper proposes a novel theoretical entrainment-impinging model of the large-scale vortex for calculating the residence enhancement coefficient (τr). The theoretical model demonstrates that the residence enhancement coefficient is a function of the cavity geometry (L, D), the vortex radius (rv), the shedding Strouhal number of the vortex (St), and the vortex/trailing edge interaction coefficient (η). Furthermore, it has been proven that the model proposed in this paper is applicable to a wide range of inflow turbulent conditions and cavity geometric configurations.

Lagrangian coherent structure analysis on transport of Acetes chinensis along coast of Lianyungang, China

Lagrangian coherent structure analysis on transport of Acetes chinensis along coast of Lianyungang, China

Analysis of the obstacle height's influence on partial cavitation flow based on the Lagrangian coherent structure

In this study, a density-modified SST k-ω model and Zwart-Gerber-Belamri cavitation model were used to numerically simulate the cavitation flow of two-dimensional NACA66 hydrofoil with obstacles of different heights, and the influence of obstacle heights on cavitation control effect was studied. The vorticity and Lagrange coherent structures (LCSs) were used to analyze the vortex structure during the cavitation development process, and the development of the vortex structure is analyzed by tracking the particle movement in the flow field. The effectiveness of Lagrange coherent structures (LCSs) for the analysis of the vortex structure in the cavitation flow was verified. The obstacles are arranged at the position where the chord length is 0.37c from the leading edge, and the heights of the obstacles are 1 mm, 1.5 mm, and 2 mm, respectively. The results show that the height of obstacles has a significant influence on the control of cavitation. The presence of obstacles can delay the beginning time of the cavitation shedding off, reduce the time ratio of the cavitation shedding off, and reduce the periodic frequency of cavitation and the maximum cavitation area. Within a certain range of obstacle height, the intensity of cavitation decreases with the increase of obstacle height, and the chaos degree of the vortex structure also decreases with the increase of obstacle height. Although the presence of obstacles does influence the intensity of cavitation development when mounted on the hydrofoil, cavitation can still extend to the trailing edge of the hydrofoil when cloud cavitation occurs.

Non-Boussinesq effect on the internal solitary wave propagation under a Lagrangian-like description for various pycnocline thickness conditions

The Dubreil–Jacotin–Long (DJL) equation is an exact model for the steady propagation of internal solitary waves (ISWs) under Boussinesq approximation. The aim of this study is to discuss the non-Boussinesq effect on ISWs initialized by DJL equation for various pycnocline thickness conditions. The simulated results show that the non-Boussinesq effect can indeed lead to instability of DJL equation, and the pycnocline thickness will affect the amplitude of re-stable ISWs and the separation process of tail waves. Moreover, in order to extract features of the non-Boussinesq instability and the location of the tail wave separation, we propose an effective analytical approach for the quasi-traveling wave problem inspired by the definition of classical Lagrangian coherent structure (LCS). The improved finite-time Lyapunov exponent with a moving window (FTLEmw) is a Lagrangian-like description with considering the prior information of physical phenomenon. For a steady ISW propagation in the Boussinesq system, the repelling and attracting FTLEmv fields are symmetrical and the field ridge behaves as a continuous spiral topology, which suggests that particles on different streamlines in the flow structure have different rotation periods, while the streamlines are not actually completely closed due to the forward propagation of the wave. During the main unstable stage of non-Boussinesq effect, the topology of separated tail wave which is disconnected from the ISW main part can be clearly distinguished. The tail wave separation locations identified by the improved FTLEmv field and the classical FTLE field are not the same, which illustrates the difference between the two perspectives of flow structure change and fluid particle motion. In addition, the leading edge, trailing edge and main part of ISWs will gradually be divided into three-part FTLEmv topologies in the thicker pycnocline conditions.

Chemodynamics of Polycyclic Aromatic Hydrocarbons and Their Alkylated and Nitrated Derivatives in the Yellow Sea and East China Sea.

The occurrence of continuously released polycyclic aromatic hydrocarbons (PAHs) in marginal seas is regulated by hydrological and biogeochemical processes; however, scarce knowledge is about their derivatives in marine environments. In this study, the dissolved and particulate PAHs and their alkylated/nitrated derivatives (A-PAHs/N-PAHs) in surface seawater of the southwestern Yellow Sea (YS) and northwestern East China Sea (ECS) during September 2022 were comprehensively discussed. Results confirm higher levels of Σ26PAHs (9.3-70 ng/L) and Σ43A-PAHs (13-76 ng/L) than Σ20N-PAHs (0.80-6.6 ng/L). The spatial heterogeneity of contaminants was regulated by substantial riverine runoff and ocean currents. Lagrangian Coherent Structure analysis further revealed the existence of a transport barrier at the shelf break of the southwestern YS where contaminants hardly crossed and tended to accumulate. The relationship between dissolved compounds and chlorophyll a indicated both biodegradation and the biological pump contributed to the depletion of PAHs and A-PAHs from surface seawater while the biological pump was the major driver for N-PAHs, despite their complicated water-particle partition behavior due to variations in physicochemical properties in the presence of nitro groups. Source identification demonstrated that pyrogenic and petrogenic sources dominated the YS and ECS, respectively, while photochemical transformations appeared more active in the YS.

Lagrangian Coherent Structures in the Mediterranean Sea: Seasonality and basin regimes

The dynamics of fluid flows give rise to robust, persistent circulation features that underpin the flow and exert strong control over the advection of water masses, either enhancing it or suppressing it, collectively known as lagrangian coherent structures. Lagrangian approaches and metrics have been shown to be better suited than eulerian ones at locating and delineating such structures and capturing the effect they have on the formation and dispersion of water masses, particularly at the smaller scales. In this paper, we use the framework of lagrangian coherent structures to analyse the ocean velocity fields over a climatological year obtained from a high-resolution eddy-resolving model in order to investigate the lagrangian regimes that affect the motion, separation and mixing of water masses in the Mediterranean Sea. The lagrangian regimes that develop in each sub-basin over the course of the year are characterised and regions of persistent lagrangian activity and coherent structure formation and presence are identified. A quantitative picture of the seasonal variability of the lagrangian coherent structure-induced horizontal mixing and vortex formation is obtained.

A quasi-objective single-buoy approach for understanding Lagrangian coherent structures and sea ice dynamics

Abstract. Sea ice drift and deformation, namely sea ice dynamics, play a significant role in atmosphere–ice–ocean coupling. Deformation patterns in sea ice can be observed over a wide range of spatial and temporal scales, though high-resolution objective quantification of these features remains difficult. In an effort to better understand local deformation of sea ice, we adapt the trajectory-stretching exponents (TSEs), quasi-objective measures of Lagrangian stretching in continuous media, to sea ice buoy data and develop a temporal analysis of TSE time series. Our work expands on previous ocean current studies that have shown TSEs provide an approximation of Lagrangian coherent structure diagnostics when only sparse trajectory data are available. As TSEs do not require multiple buoys, we find they have an expanded range of use when compared with traditional Eulerian buoy-array deformation metrics and provide local-stretching information below the length scales possible when averaging over buoy arrays. We verify the ability of TSEs to temporally and spatially identify dynamic features for three different sea ice datasets. The ability of TSEs to quantify trajectory stretching is verified by concurrent ice fracture in buoy neighborhoods ranging from tens to hundreds of kilometers in diameter, as well as the temporal concurrence of significant storm events.

Mapping the shape and dimension of three-dimensional Lagrangian coherent structures and invariant manifolds

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

Generic Energy Function To Predict Particle Positions In 4D-PTV

Recent developments in time-resolved Particle Tracking Velocimetry (4D-PTV) consistently improved tracking accuracy and robustness by introducing predictive algorithms such as Shake-The-Box (STB). We propose a generic nondimensional energy function to predict particle positions by leveraging additional information from coherent neighbour motions. The proposed energy function is a combination of position history, neighbour velocity and acceleration terms. Coherent neighbours are quantified by locally utilising Finite Time Lyapunov Exponent (FTLE) as an objective Lagrangian Coherent Structure (LCS) diagnostic method. Synthetic analysis of the wake behind a smooth cylinder at Reynolds number equal to 3900 showed that the optimal solution of the minimised energy function could be modelled as a function of the measurement uncertainties. The model was assessed with the 2D homogeneous isotropic turbulent flow (HIT) as well and was found to be case and Reynolds independent. Results of the 4D-PTV experimental study of the same wake flow configurations are reported. We compared predicted positions with the optimised final positions of STB. It was found that the generic energy function succeeded in estimating particle positions with minimum deviation to the optimised positions in comparison with other techniques.

Applying finite-time lyapunov exponent to study the tidal dispersion on oil spill trajectory in Burrard Inlet

This study used a combination of Finite-Time Lyapunov Exponent (FTLE) values, residual currents, and tidal excursion lengths to systematically investigate the effects of tidal dispersion on oil spill trajectories in Burrard Inlet, BC, Canada, which is a tidally dominated estuary. The FTLE analysis results showed that tidal type and tidal phase significantly influenced the FTLE fields because the flow structure and the location of saddle points varied as a function of the tidal type and tidal phase. Some transport barriers formed in the Inner Harbour, which blocked the water exchange between the western and eastern parts of the inlet. Moreover, tidal mixing in the wider regions of Burrard Inlet (i.e., the western Outer Harbour) was relatively weak than in the narrower areas (i.e., First Narrows and Second Narrows). The observations from the FTLE analysis agreed well with the residual currents and tidal excursion results. The drifter trajectories were consistent with the Lagrangian coherent structure lines extracted from the FTLE analysis. To verify the tidal dispersion impact on an oil spill trajectory in the inlet, the FTLE fields were compared with a real oil spill that occurred in Burrard Inlet in 2015 (the M/V Marathassa oil spill). The FTLE fields reasonably explained the spilled oil’s trajectories from the real event. In addition, a set of stochastic oil spill models were run in this study and found that the FTLE analysis was a reliable tool for oil spill tracking. Overall, the FTLE method would be a valuable addition to practical oil spill response planning.

Study on the vidcous flow around foils with a multi-resolution smooth particle hydrodynamics method

Considering the real viscosity of the fluid, the multi-resolution smooth particle hydrodynamics method is applied to numerically simulate and verify the flow around the foils at different Reynold numbers. In the framework of SPH, the Lagrangian coherent structure can be easily visualized, thus the details of the vortex can be reproduced. In order to overcome the SPH numerical instability when simulating the flow around airfoil at high Reynolds number, the tension instability control and the particle shifting techniques are adopted, thus the numerical accuracy of the pressure and velocity fields are much improved. The analysis of the flow around the foil shows that the SPH method can accurately calculate the velocity and pressure fields, and perform good force monitoring and prediction. In addition, compared to the traditional vortex capturing method based on Euler velocity field, the Lagrangian coherent structure can also effectively reveal the characteristics and the evolution of vortex. The study indicates that the SPH method owns distinguished advantages in the simulation of fluid dynamics problems considering the real viscous boundary layer and vortex motion. In addition, the investigation of the flow around flapping foil demonstrates the capability of the SPH method for simulation with moving boundaries.

Lagrangian Coherent Structure Analysis on the Vegetated Compound Channel with Numerical Simulation

Natural channels often consist of a mainstream near their thalwegs and shallow vegetated areas near shores. The compounded and partially vegetated cross-sections play a significantly role in determining the hydrodynamic characteristics of a channel. By employing the Lagrangian Coherent Structure (LCS) analysis, the present work unravels the effect of vegetation and geometry on the hydrodynamic interactions between mainstreams with the various depths and vegetated shallow areas. The LCS method is the concept of dynamical system analyses, which is determined by the finite-time Lyapunov exponents (FTLE) field of fluid particles. It enables to overcome the limitations of using the particle tracking method in cost and time for simulations. Since the LCSs represent material surfaces or asymptotic lines which particles approach, but do not pass through, they match well with the trajectories of particles or materials obtained by solving particle motion equations. Therefore, the temporal and spatial developments of the interfacial layers could be investigated by using the FTLE. As the difference of depth becomes appreciable, the values of FTLE are relatively larger farther from the vegetated area. It implies that the interfacial layer becomes wider with the larger size of vortex produced by the differences of velocities between the mainstreams and the vegetated areas. In other words, as depth differences become large, materials and momentum can be spread from the vegetated area to or collected from a wider area of the mainstream.

FAST IDENTIFICATION OF THE HYPERBOLIC LAGRANGIAN COHERENT STRUCTURES IN TWO-DIMENSIONAL FLOWS BASED ON THE EULERIAN-TYPE ALGORITHMS

<abstract> Based on the so-called mixing-based partition, we propose an efficient Eulerian algorithm to identify the hyperbolic Lagrangian coherent structure (LCS) of any given two-dimensional (2D) flow fields, which is framework independent. To extract the required LCS, the proposed algorithm only needs to solve one single partial differential equation. Moreover, data is only required at mesh points in the implementation of the proposed algorithm. In contrast, traditional Lagrangian ray tracing approach needs to solve a system consisting of two ordinary differential equations together with a line integral along the particular particle trajectory. Furthermore, if the velocity data is only available at mesh points, the Lagrangian approach needs to implement interpolation to obtain the velocity and also the velocity gradient at non-mesh points along the particle trajectory taking off from each mesh point, which could be quite time-consuming. Based on the doubling technique, we also propose an efficient iterative Eulerian-type algorithm to identify the longtime LCS for 2D periodic flows. Numerical examples are provided to confirm the accuracy, efficiency and effectiveness of the proposed Eulerian algorithms. </abstract>

Vortex structures evolution in supersonic mixing layers with different inlet Reynolds numbers based on the Lagrangian method

The supersonic mixing layer is a typical flow structure in an aircraft’s optical seeker. The free incoming flow and cooling jet form a mixing layer through shearing. In most cases, the flow has become a fully developed turbulent due to complex factors during flight. This paper mainly focuses on the influence of the inlet Reynolds number on the development of the supersonic mixing layer. The rescaling/recycle method is used to generate turbulence at the inlet of the mixing layer. The results show that when the turbulence of the high-speed flow is strong, the transition position of the mixing layer will be advanced and the mixedness will be enhanced. The mixing efficiency is high at the position corresponding to the vortex in the mixing layer. Finally, the Lagrangian coherent structure and particle tracking methods are used to analyze different mixing processes, and the entrainment characteristics of the vortex are studied.

Lagrangian coherent structures and material transport in unsteady flow of vertical-axis turbine wakes

In this study, the unsteady phenomena of vertical-axis turbine wake flow were analyzed in two rotational speed conditions from a Lagrangian perspective. The wake flow field was obtained using particle image velocimetry. The spectral proper orthogonal decomposition method was used to decompose the wake modes with different frequencies. The Lagrangian coherent structure theory based on the finite-time Lyapunov exponent was used to study the material transport and mixing in the wake evolution process. The results show that the Lagrangian coherent structures can capture the boundary of the wake vortex. At a low rotation speed, a vorticity–flux window is found, through which the fluid is transported to the rotation-induced structures. At a high rotation speed, the increasing material transport between the separation bubbles and the incoming flow eventually leads to shedding of the large-scale vortex structures, accompanied by a decrease in the length of the separation zone. The material transport and mixing revealed in the unsteady flow of turbine wakes have significance in guiding the study of the flow control mechanism.

Forced convection heat transfer around a circular cylinder in laminar flow: An insight from Lagrangian coherent structures

The Lagrangian coherent structure (LCS) method is introduced to the convection heat transfer problem, and the forced convection heat transfer around a circular cylinder in laminar flow regime is analyzed from the Lagrangian viewpoint. First, the mechanics model of forced convection heat transfer around a circular cylinder is introduced along with the mathematical formulations. Subsequently, an implicit flow solver based on the characteristic-based split finite element method and the dual-stepping method is proposed to solve the flow and heat transfer equations, and the computation of LCS employing the finite-time Lyapunov exponent is discussed in detail. The accuracy and stability of the flow solver are examined carefully utilizing the data reported in the literature, and the grid-independence test is conducted. Finally, the mass and energy transport features around the circular cylinder at Pr (Prandtl number) = 0.7 and Re (Reynolds number) = 20–180 are investigated by relating the instantaneous thermal and vorticity patterns, lift coefficient, drag coefficient, and Nusselt number to the LCSs in the flow field. The results obtained may improve the existing understanding of forced convection heat transfer around a circular cylinder and pave the road for the application of LCSs in convection heat transfer problems.

Motion-shape-based deep learning approach for divergence behavior detection in high-density crowd

We propose a novel method of abnormal crowd behavior detection in surveillance videos. Mainly, our work focuses on detecting crowd divergence behavior that can lead to serious disasters like a stampede. We introduce a notion of physically capturing motion in the form of images and classify crowd behavior using a convolution neural network (CNN) trained on motion-shape images (MSIs). First, the optical flow (OPF) is computed, and finite-time Lyapunov exponent (FTLE) field is obtained by integrating OPF. Lagrangian coherent structure (LCS) in the FTLE field represents crowd-dominant motion. A ridge extraction scheme is proposed for the conversion of LCS-to-grayscale MSIs. Lastly, a supervised training approach is utilized with CNN to predict normal or divergence behavior for any unknown image. We test our method on six real-world low- as well as high-density crowd datasets and compare performance with state-of-the-art methods. Experimental results show that our method is not only robust for any type of scene but also outperform existing state-of-the-art methods in terms of accuracy. We also propose a divergence localization method that not only identifies divergence starting (source) points but also comes with a new feature of generating a ‘localization mask’ around the diverging crowd showing the size of divergence. Finally, we also introduce two new datasets containing videos of crowd normal and divergence behaviors at the high density.