Finite Size Lyapunov Exponent Research Articles

Assessing the Impact of Data-resolution On Ocean Frontal Characteristics

AbstractEasy access to and advances in satellite remote sensing data has enabled enhanced analysis of ocean fronts, physical and ecologically important areas where water masses converge. Recent development of higher-resolution satellite imagery to detect ocean fronts provides the potential to better capture patterns and trends of ocean change and improve modelling and prediction efforts. This study examines the relationship between satellite data spatial resolution and its influence on the quantification of frontal characteristics, frontal quantity, length, strength and density. We also examine the relationship between Finite-Size Lyapunov Exponents and image resolution. We found higher spatial resolution leads to increased frontal quantity and decreased frontal length. Also, both strength and spatial density of fronts differ at various resolutions. The Finite-Size Lyapunov Exponent value does not change significantly with resolution. Knowledge of the impact of resolution on the quantification of frontal characteristics is crucial as it enables the exploration of novel experimental design to further facilitate the development of improved parameterization and uncertainties in ocean modelling/studies.

Offshore Plankton Blooms via Mesoscale and Sub‐Mesoscale Interactions With a Western Boundary Current

AbstractWe investigate mesoscale circulations in an oligotrophic western boundary current, the East Australian Current, during a sustained offshore plankton bloom. Using a series of high resolution hydrographic sections taken a few days apart, supplemented with in situ samples of nutrients, satellite and long‐term mooring measurements, we describe a dynamic situation by which the East Australian Current's velocity core and associated front, interacts with a mesoscale eddy, migrating zonally by approximately 100 km over the course of 10 days. This interaction between the boundary current and mesoscale eddy field occurred concurrently with a sustained offshore plankton bloom. Sub‐mesoscale upwelling motions on the inshore flank of the boundary current core coincides with increased nutrient and plankton concentrations in the near surface. Calculations based on the quasi‐geostrophic omega equation and finite size lyapunov exponents suggest that these vertical motion arise from the interaction of the mesoscale with the East Australian Current. Frontolysis (the destruction of horizontal buoyancy gradients) leads to a thermally indirect ageostrophic secondary circulation that has the potential to supply nutrients to the near surface ocean. Using satellite data, we investigate the mesoscale conditions associated with all offshore phytoplankton blooms identified by an automated method, finding similar mesoscale patterns to those observed during the field campaign. We conclude that the interaction between the East Australian Current and mesoscale eddies is a recurrent catalyst for the complex sub‐mesoscale dynamics we observed, and is likely a fundamental processes in driving offshore biological productivity in the region.

Coastal current convergence structures in the Bay of Biscay from optimized high-frequency radar and satellite data

The southeastern Bay of Biscay has been described as a “dead end” for floating marine litter, often accumulating along small-scale linear streaks. Coastal Current Convergence Structures (CCS), often associated with vertical motions at river plume edges, estuarine fronts, or other physical processes, can be at the origin of the accumulation. Understanding the formation of CCS and their role in the transport of marine litter is essential to better quantify and to help mitigate marine litter pollution. The Lagrangian framework, used to estimate the absolute dispersion, and the finite-size Lyapunov exponents (FSLE), have proved very effective for identifying CCS in the current velocity field. However, the quality of CCS identification depends strongly on the Eulerian fields. Two surface current velocity data sets were used in the analysis: the remotely sensed velocities from the EuskOOS High-Frequency Radar (HFR) network and velocities from three-dimensional model outputs. They were complemented by drifting buoy velocity measurements. An optimization method, involving the fusion of drifting buoys and HFR velocities is proposed to better reconstruct the fine-scale structure of the current velocity field. Merging these two sources of velocity data reduced the mean Lagrangian error and the Root Mean Square Error (RMSE) by 50 % and 30 % respectively, significantly improving velocity reconstruction. FSLE ridgelines obtained from the Lagrangian analysis of optimized velocities were compared with remotely sensed concentrations of Chlorophyll-a. It was shown that ridgelines control the spatial distribution of phytoplankton. They fundamentally represent the CCS which can potentially affect marine litter aggregation. Analysis of the absolute dispersion revealed large stirring in the alongshore direction which was also confirmed by spatial distribution of FSLE ridgelines. The alignment between FSLE ridgelines and patterns of high Chlorophyll-a concentration was observed, often determining the limits of river plume expansion in the study area.

Study of upwelling and mixing process in the Somali coastal region using satellite and numerical model observations: A Lagrangian approach

During the summer monsoon, the Somali region undergoes a significant upwelling phenomenon that enhances plankton productivity, thereby benefiting fisheries. Wind and coastal dynamics initially drive this upwelling, but eventually, eddy flows influence it. Our study explores the interplay between ocean currents, eddies, and chlorophyll-a concentrations using the backward Finite-Size Lyapunov Exponents (bFSLEs) technique. We also delve into the specific role of Ekman transport in distributing chlorophyll-a across the region. The Great Whirl (GW), an anticyclonic eddy, predominantly causes strong downwelling, interrupting the summer monsoon upwelling along the Somali coast longitudinally. Despite the GW's significant impact on moving upwelled water offshore, the influence of downwelling diminishes northward. As a result, the northern Somali coast, especially around 9°N and 10°N, showcases the most extensive offshore upwelling, reaching as far as 55°E. Our findings highlight a robust connection between chlorophyll-a levels and oceanic dynamics, influenced by both currents and eddies, as evidenced by bFSLEs, and by cross-shore Ekman transport, particularly within chlorophyll-a concentrations ranging from 0.2 to 0.6 mg m−3. The data suggests that Ekman transport-induced upwelling primarily drives coastal phytoplankton biomass. Furthermore, bFSLEs analysis underlines the supportive role of ocean currents and eddies in the offshore distribution of chlorophyll-a, especially near the coast. Further examination of lagged correlations reveals a temporal lag between peak concentrations of chlorophyll-a and Ekman transport; the lag increases offshore and is at least 9 days near the coast.

Relative dispersion and kinematic properties of the coastal submesoscale circulation in the southeastern Ligurian Sea

Abstract. An array of Lagrangian instruments (more than 100 drifters and a profiling float) were deployed for several days in the coastal waters of the southeastern Ligurian Sea to characterize the near-surface circulation at the submesoscale (< 10 km). The drifters were trapped in an offshore-flowing filament and a cyclonic eddy that developed at the southwestern extremity of the filament. Drifter velocities are used to estimate differential kinematic properties (DKPs) and the relative dispersion of the near-surface currents on scales as small as 100 m. The maximum drifter speed is ∼ 50 cm s−1. The DKPs within the cluster exhibit considerable spatial and temporal variability, with absolute values reaching the order of magnitude of the local inertial frequency. Vorticity prevails in the core of the cyclonic eddy, while strain is dominant at the outer edge of the eddy. Significant convergence was also found in the southwestern flow of the filament. The initial relative dispersion on small scales (100–200 m) is directly related to some of the DKPs (e.g., divergence, strain and instantaneous rate of separation). The mean squared separation distance (MSSD) grows exponentially with time, and the finite-size Lyapunov exponent (FSLE) is independent of scale. After 5–10 h of drift or for initial separations greater than 500 m, the MSSD and FSLE show smaller relative dispersion that decreases slightly with scale.

Influence of ocean dynamics on Cuvier’s beaked whale (Ziphius cavirostris): Presence and their prey

Cuvier’s beaked whales (Ziphius cavirostris; Zc) dive below 1000 m to forage predominantly on deep-sea squids. The impact of oceanographic conditions on Zc prey and the effects of anthropogenic sounds on Zc foraging remain unclear. Zc prey distribution is patchy and potentially influenced by mesoscale oceanographic features interacting with local bathymetry. Favorable environmental conditions lead to aggregations of lower trophic level species that attract squid, and potentially increase Zc presence. We collected simultaneous active and passive acoustic data in Tanner Basin, a southern California submarine canyon, intermittently between 2017 and 2022. Passive acoustic data revealed Zc were present throughout all years, with lower presence during late summer and fall. Subsurface Zc 3D track reconstructions indicated a preference for one side of the canyon, where Zc foraged within a layer of potential prey, detected from active acoustic data. Prey concentrations, quantified as volume backscatter strength below ∼1150 m depth, exhibited an increase following the passage of mesoscale features, which were measured using Finite-size Lyapunov Exponents (FSLE). These prey increases drove immediate increases in Zc presence. Understanding potential physical drivers of Zc predator-prey dynamics is a critical first step for assessing natural variability and potential impacts of sonar use on Zc foraging behavior.

Ageostrophic Contribution by the Wind and Waves Induced Flow to the Lateral Stirring in the Mediterranean Sea

AbstractWe study the impact of the Ekman currents and Stokes drift on the surface horizontal transport and mixing properties of the Mediterranean Sea. Finite Size Lyapunov Exponents (FSLE) at the ocean surface are computed over the whole basin using 25 years of satellite altimetry derived geostrophic currents, 10‐m wind velocity and wave fields. We find that the transport pathways unveiled by the geostrophic Lagrangian Coherent Structures (LCS) are significantly modified by the ageostrophic currents (i.e., Ekman and Stokes induced velocities), often leading to a decrease of the retention capacity of the eddies. An exhaustive assessment of the regional and temporal variability of the FSLE shows an increase of the horizontal mixing processes, due to the ageostrophic component, up to 37% in regions such as the Gulf of Lion or the Aegean Sea, during the seasons where wind and waves are intense and persistent. Positive trends in the total FSLE (up to 1.2% of the value of FSLE per year in some regions) suggest that Mediterranean Sea has experienced a significant increase in the intensity of the mixing processes over the last decades. Ageostrophic features are considered to play a role in determining the properties of the relative dispersion. Through the analysis of the Lagrangian Anisotropy Index (LAI) using virtual and real pair of drifters, we observe that the particle dispersion is mainly dominated by the zonal flow, and that the ageostrophic currents induce meridional dispersion, particularly in regions where wind and wave are intensified.

A Lagrangian study of summer upwelling along the Uruguayan coast

This study focuses on the surface transport in the Uruguayan coastal zone during summertime under favorable wind upwelling conditions. The surface transport in the area was investigated for the first time with the application of a Lagrangian technique computing the backward-in-time Finite Size Lyapunov Exponents (FSLE) using the current velocity field derived from a high resolution numerical model (CROCO, 1/36° horizontal grid). Intense summer upwellings were studied focusing on surface stirring characterized by FSLE temporal averages. Subsequently, the analysis of the attracting FSLE field maxima allowed the detection of Lagrangian Coherent Structures (LCS), providing information on the location of the upwelling frontal regions and on the transport patterns associated with the flow. The FSLE analysis during summertime revealed an intense mixing activity concentrated in the oceanic coastal area, concordant with the location of the summer climatological upwelling, while during the most intense upwelling dates an increase in the mixing activity was found in the estuarine coastal region. When concentrating on the FSLE maxima during intense upwelling dates, we identified attractive LCS in the estuarine region locating retention fronts. The identified structures' location coincided with the frontal regions observed from both, simulated SST and satellite Chlorophyll-a distribution, highlighting the Lagrangian approximation's importance in locating retention fronts with different origins. Finally, the LCS evolution analyzed for two particular upwelling events, together with the simulated dispersion of synthetic Lagrangian particles, evidenced the importance of this approximation to study the transport processes in the area.

Relación entre frentes de convergencia y localización de la flota pesquera durante la ocurrencia de surgencia costera en Uruguay

The Uruguayan coast presents upwelling events during the summer, its implications on the distribution, transport of nutrients and organisms; and their effects on the entire food web, have not been previously analyzed. This work seeks to understand the relationship between the upwelling events and the location of the Uruguayan coastal industrial fleet (B Category). Finite Size Lyapunov Exponents (FSLE) were computed from surface current velocity fields. The temporal averages of these exponents indicate the mixing intensity and presented higher values over the region of more intense upwelling. On the other hand, the maximum FSLE values identify convergence fronts. The fronts identified by this methodology correspond to the turbidity front and close to Montevideo, and temperature fronts directly associated with the upwelling event. During the event, these fronts migrated from the coast, to the southwest and offshore. The location of the frontal regions detected was compared with the satellite position (VMS) of B Category coastal vessels of the Uruguayan fishing fleet. In general, the vessels were positioned on the frontal zones, suggesting a relationship between retention processes and the location of target species; and evidencing the need to deepen the study of these processes.

Generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: Some exact results.

We consider the one-dimensional Schrödinger equationwith a random potential and study the cumulant generating function of the logarithm of the wave function ψ(x), known in the literature as the "generalized Lyapunov exponent"; this is tantamount to studying the statistics of the so-called "finite-size Lyapunov exponent." The problem reduces to that of finding the leading eigenvalue of a certain nonrandom non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus on the case of Cauchy disorder, for which we derive a secular equationfor the generalized Lyapunov exponent. Analytical expressions for the first four cumulants of ln|ψ(x)| for arbitrary energy and disorder are deduced. In the universal (weak-disorder and high-energy) regime, we obtain simple asymptotic expressions for the generalized Lyapunov exponent and for all the cumulants. The large deviation function controlling the distribution of ln|ψ(x)| is also obtained in several limits. As an application, we show that, for a disordered region of size L, the distribution W_{L} of the conductance g exhibits the power-law behavior W_{L}(g)∼g^{-1/2} as g→0.

Assessment of relative dispersion in the Gulf of Tonkin using numerical modeling and HF radar observations of surface currents

Particle pair statistics from synthetic drifter trajectories reconstructed from realistic, high-resolution numerical simulations (SYMPHONIE model) and HF radar velocity measurements are used to investigate the dispersion properties in the Gulf of Tonkin (GoT). This study takes an approach based on two-particles statistics providing the relative dispersion, relative diffusivity and Finite Size Lyapunov Exponent (FSLE) estimates. In the GoT, the relative dispersion follows the predictions from the theory of two-dimensional turbulence with two inertial subranges identified in the kinetic energy spectrum with the spectral slopes −5/3 and −3. The time evolution of dispersion shows an exponential growth during 5–8 days, followed by a power law regime during the next 6–20 days. Fixed-length indicators from the relative diffusivity and the FSLE reveal a local dispersion at large and intermediate scales (above Rossby radius of deformation) and non-local dispersion in sub-mesoscale range (below Rossby radius of deformation). The effect of river runoff on the local hydrodynamics and dispersion processes is assessed using the numerical model simulations without river discharge. The results show that in the model, the river plume, when present, highly impacts the Lagrangian statistics. High gradients of buoyancy reinforce the sub-mesoscale circulation in a large region along the Vietnamese coast and modify the scales and intensity of turbulent dispersion. However, a clear change of dispersion regime in the sub-mesoscale range is not identified, suggesting that the mesoscale circulation in the GoT largely governs particle spreading even at small scale.

A Study of Mixing and Biological Activity in the North Indian Ocean Using Finite Size Lyapunov Exponents

A Study of Mixing and Biological Activity in the North Indian Ocean Using Finite Size Lyapunov Exponents

Monthly to Decadal Variability of Mesoscale Stirring in the California Current System: Links to Upwelling, Climate Forcing, and Chlorophyll Transport

AbstractThe California Current System (CCS) is a highly productive eastern boundary upwelling system. Cross‐shore transport is driven by horizontal stirring from mesoscale (and submesoscale) eddies, fronts, and filaments, which shift surface productivity away from the narrow upwelling zone inshore. Using an unprecedented 25‐year daily record of finite size Lyapunov exponents (FSLEs), we characterize the spatial and temporal patterns of mesoscale stirring in the entire CCS and quantify correlations of FSLEs with satellite‐measured chlorophyll a and climate indices. The annual cycle of mesoscale stirring is highly correlated with seasonal upwelling in the CCS, with bands of intense FSLE propagating to the west during the upwelling season. Annual cycles and interannual variability in FSLE intensity were most prominent to the north and inshore of the CCS, and weaker to the south and offshore. Interannual fluctuations in mesoscale stirring showed 3–12 month lagged responses to climate indices, with the CCS oscillating out of phase with waters west and north. Chlorophyll a is positively correlated with FSLE intensity in a meridional band 200–600 km offshore, consistent with the hypothesis that increased horizontal stirring increases the export of chlorophyll a to offshore waters. When mesoscale stirring intensifies during negative phases of the Pacific Decadal Oscillation and Ocean Niño index, we predict greater offshore transport of organic carbon. Our analyses underscore the predictability of annual and interannual cycles of mesoscale stirring and chlorophyll a fluxes in the CCS and their links with climate indices.

Experimental study of chaotic flow and mixing of Newtonian fluid in a rotating arc-wall mixer

In order to mix highly viscous fluids with minimal energetic input, a new active in-line mixer has been developed in our previous study (El Omari et al., Phys. Rev. Fluids, 2021 [18]). Based on the principle of mixing by chaotic advection, we present in this paper an experimental characterisation of chaotic mixing of Newtonian fluid flows, at low Reynolds number. First, the flow is characterized using velocity field measurements. Distinct flow topologies are detected in the flow. The trajectories of the fluid particles are computed and show the generation of complex structures in the flow. Residence time distributions reveal that the fluid particles spend more time in the mixer under favorable controlling conditions. Finite size Lyapunov exponents are calculated and indicate that the flow is more chaotic for these conditions. Next, the mixing patterns are visualized and showed that the mixing is more homogeneous under the same favorable conditions for which the fluid particles are sufficiently subjected to the stretching and folding mechanism.

The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.

A Deep Water Dispersion Experiment in the Gulf of Mexico

AbstractThe Deep Water Horizon oil spill dramatically impacted the Gulf of Mexico from the seafloor to the surface. While dispersion of contaminants at the surface has been extensively studied, little is known about deep water dispersion properties. This study describes the results of the Deep Water Dispersion Experiment (DWDE), which consisted of the release of surface drifters and acoustically tracked RAFOS floats drifting at 300 and 1,500 dbar in the Gulf of Mexico. We show that surface diffusivity is elevated and decreases with depth: on average, diffusivity at 1,500 dbar is 5 times smaller than at the surface, suggesting that the dispersion of contaminants at depth is a significantly slower process than at the surface. This study also examines the turbulent regimes driving the dispersion, although conflicting evidences and large uncertainties do not allow definitive conclusions. At all depths, while the growth of dispersion and kurtosis with time supports the possibility of an exponential regime at very short time scales, indicating that early dispersion is nonlocal, finite size Lyapunov exponents support the hypothesis of local dispersion, suggesting that eddies of size comparable to the initial separation (6 km), may dominate the early dispersion. At longer time scales, the quadratic growth of dispersion is indicative of a ballistic regime, where a mean shear flow would be the dominating process. Examination of the along‐ and across‐bathymetry components of float velocities supports the idea that boundary currents could be the source for this shear dispersion.

Analysis of mixing structures in the Adriatic Sea using finite-size Lyapunov exponents

In this paper, we report on the mixing structures and transport properties of the Adriatic Sea surface, as a semi-enclosed basin of the Mediterranean Sea, from October 2006 until December 2011. Lagrangian transport models were used to simulate synthetic trajectories from the mean flow fields obtained by the Massachusetts Institute of Technology general circulation model implemented for the Adriatic. We examine the dispersion properties of numerical pair particles, through the calculation of time-averaged finite-size Lyapunov exponents (FSLEs) in the Adriatic Sea during selected months in each year. The results show the significant effects of river runoff and wind forcing, especially the Bora wind field, on the mixing activities of numerical pair particles by the generation of vortices, which appear as a tangle of filaments on the FSLE maps. The stretch/compression lines, which contain high values of the FSLEs, work as robust transport barriers, most having been detected along boundary currents on the eastern and western flanks of the Adriatic, particularly during winter. Numerical experiments have indicated that stable flows, with less mixing activity, occur in the northern part of the Adriatic in June and September of each year, while the Southern Adriatic Pit has flows with larger seasonal fluctuations and high values of eddy kinetic energy because of the influence of wind and energetic currents entering from the Ionian Sea.

Observing and quantifying ocean flow properties using drifters with drogues at different depths

AbstractThis paper presents analyses of drifters with drogues at different depths – 1, 10, 30, 50 m – that were deployed in the Mediterranean Sea to investigate frontal subduction and upwelling. Drifter trajectories were used to estimate divergence, vorticity, vertical velocity, and finite-size Lyapunov exponents (FTLEs), and to investigate the balance of terms in the vorticity equation. The divergence and vorticity are O(f) and change sign along trajectories. Vertical velocity is O(1 mm/s), increases with depth, indicates predominant upwelling with isolated downwelling events, and sometimes changes sign between 1 and 50 m. Vortex stretching is one of, but not the only, significant term in the vorticity balance. 2D FTLEs are 2 × 10−51/s after 1 day, twice larger than in a 400-m-resolution numerical model. 3D FTLEs are 50% larger than 2D FTLEs and are dominated by the vertical shear of horizontal velocity. Bootstrapping suggests uncertainty levels of ~10% of the time-mean absolute values for divergence and vorticity. Analysis of simulated drifters in a model suggests that drifter-based estimates of divergence and vorticity are close to the Eulerian model estimates, except when drifters get aligned into long filaments. Drifter-based vertical velocity is close to the Eulerian model estimates at 1 m but differs at deeper depths. The errors in the vertical velocity are largely due to the lateral separation between drifters at different depths, and partially due to only measuring at 4 depths. Overall, this paper demonstrates how drifters, heretofore restricted to 2D near-surface observations, can be used to learn about 3D flow properties throughout the upper layer of the water column.

Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson localization models

For the 2D matrix Langevin dynamics that correspond to the continuous-time limit of the products of some 2 × 2 random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can thus be analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and empirical current of the Riccati process, and one then performs the appropriate Euler–Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker–Planck operator. The associated conditioned process obtained via the generalization of Doob’s h-transform allows one to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson localization models are described in order to explicitly obtain the first cumulants of the finite-size Lyapunov exponent.

Predicting Skipjack Tuna Fishing Grounds in the Western and Central Pacific Ocean Based on High-Spatial-Temporal-Resolution Satellite Data

Skipjack tuna are the most abundant commercial species in Taiwan’s pelagic purse seine fisheries. However, the rapidly changing marine environment increases the challenge of locating target fish in the vast ocean. The aim of this study was to identify the potential fishing grounds of skipjack tuna in the Western and Central Pacific Ocean (WCPO). The fishing grounds of skipjack tuna were simulated using the habitat suitability index (HSI) on the basis of global fishing activities and remote sensing data from 2012 to 2015. The selected environmental factors included sea surface temperature and front, sea surface height, sea surface salinity, mixed layer depth, chlorophyll a concentration, and finite-size Lyapunov exponents. The final input factors were selected according to their percentage contribution to the total efforts. Overall, 68.3% of global datasets and 35.7% of Taiwanese logbooks’ fishing spots were recorded within 5 km of suitable habitat in the daily field. Moreover, 94.9% and 79.6% of global and Taiwan data, respectively, were identified within 50 km of suitable habitat. Our results showed that the model performed well in fitting daily forecast and actual fishing position data. Further, results from this study could benefit habitat monitoring and contribute to managing sustainable fisheries for skipjack tuna by providing wide spatial coverage information on habitat variation.