Finite Size Lyapunov Exponent Research Articles

Boundaries of the Peruvian oxygen minimum zone shaped by coherent mesoscale dynamics

Oxygen minimum zones exert important controls over ocean biogeochemistry. Lagrangian modelling demonstrates that the mean positions of mesoscale eddies delimit the boundaries of the Peruvian oxygen minimum zone. Dissolved oxygen in sea water affects marine habitats and biogeochemical cycles1,2,3. Oceanic zones with oxygen deficits represent 7% of the volume and 8% of the area of the oceans4, and are thought to be expanding4,5. One of the most pronounced lies in the region off Peru, where mesoscale activity in the form of fronts and eddies is strong. Here, we study the dynamics of the Peruvian oxygen minimum zone in a Lagrangian framework, using a coupled physical–biogeochemical numerical model and finite-size Lyapunov exponent fields, to evaluate the role of mesoscale activity. We find that, at depths between 380 and 600 m, mesoscale structures have two distinct roles. First, their mean positions and paths delimit and maintain the oxygen minimum zone boundaries. Second, their high-frequency fluctuations inject oxygen across the oxygen minimum zone boundaries and eddy fluxes are one order of magnitude higher than mean oxygen fluxes. We conclude that these eddy fluxes contribute to the ventilation of the Peruvian oxygen minimum zone.

Predictability of the Northeast Monsoon Forecasts by the Educational Global Climate Model

The predictability by an atmospheric prediction model is determined by the uncertainties in the initial condition and the imperfection of the model. It is difficult to provide accurate weather prediction and determines the predictability of a model. Atmospheric prediction model efficiency is obtained from the analysis of predictability measurement. Five existing predictability measurements; Lyapunov exponent, finite size Lyapunov exponent, finite time Lyapunov exponent, local Lyapunov exponent and largest Lyapunov exponent are used to measure predictability of the northeast monsoon (winter monsoon) by the Educational Global Climate Model (EdGCM) and to test sensitivity of the model to small initial perturbations. The EdGCM is run for 142-year predictions from the year 1958 to 2100. However, only the outputs of geopotential height at 500hPa of December from 2012 to 2100 are used for predictability measurement. The results show that the EdGCM predictability for the northeast monsoon forecast is about 120 years.

Variation of flow properties during a collision event of two mesoscale eddies in the Gulf Stream region from numerical simulation

The temporal variation of the flow structure and consequent mixing process during the collision of two counter-rotating mesoscale eddies are investigated by analyzing the HYbrid Coordinate Ocean Model simulation for the Gulf Stream region using Eulerian parameters—Okubo-Weiss parameters and horizontal kinetic energy (KE)—and Lagrangian parameters—finite-size Lyapunov exponent (FSLE) and relative dispersion coefficients (Kr). During the collision process, a transport barrier constructed by FSLE ridges develops between the two eddies and hyperbolic points are formed at both ends of the barrier. High values of the shear components of strain (> mean + standard deviation) are observed around the hyperbolic points, indicating possible deformation of the eddy. The magnitudes of spatially averaged KE and FSLE values increase (~20% and ~25%, respectively) during the collision as the flows around the main eddy become more energetic and dispersive. The Eulerian measures—the relative vorticity and the shear components of strain—show different temporal evolutions. The former does not significantly vary (~3%) while the latter has a peak value (~34%) at the time of maximum impact of the collision. In contrast, the Lagrangian measures show a similar pattern of temporal variations as both FSLE and Kr values generally increase (~25% and ~35%, respectively) during the collision, which indicates increased mixing due to the collision.

Fickian dispersion is anomalous

The thesis put forward here is that the occurrence of Fickian dispersion in geophysical settings is a rare event and consequently should be labeled as anomalous. What people classically call anomalous is really the norm. In a Lagrangian setting, a process with mean square displacement which is proportional to time is generally labeled as Fickian dispersion. With a number of counter examples we show why this definition is fraught with difficulty. In a related discussion, we show an infinite second moment does not necessarily imply the process is super dispersive. By employing a rigorous mathematical definition of Fickian dispersion we illustrate why it is so hard to find a Fickian process. We go on to employ a number of renormalization group approaches to classify non-Fickian dispersive behavior. Scaling laws for the probability density function for a dispersive process, the distribution for the first passage times, the mean first passage time, and the finite-size Lyapunov exponent are presented for fixed points of both deterministic and stochastic renormalization group operators. The fixed points of the renormalization group operators are p-self-similar processes. A generalized renormalization group operator is introduced whose fixed points form a set of generalized self-similar processes. Power-law clocks are introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed.

수치모델 해상도가 중규모 와동 근처의 난류구조에 미치는 영향

Three-dimensional structures of large ocean rings in the Gulf Stream region are investigated using the HYbrid Coordinate Ocean Model (HYCOM). Numerically simulated flow structures around four selected cyclonic and anticyclonic rings are compared with two different horizontal resolutions: 1/12° and 1/48° . The vertical distributions of Lagrangian Coherent Structures (LCSs) are analyzed using Finite Size Lyapunov Exponent (FSLE) and Okubo-Weiss parameters (OW). Curtain-shaped FSLE ridges are found in all four rings with extensions of surface ridges throughout the water columns, indicating that horizontal stirring is dominant over vertical motions. Near the high-resolution rings, many small-scale flow structures with size O(1~10) km are observed while these features are rarely found near the low-resolution rings. These small-scale structures affect the flow pattern around the rings as flow particles move more randomly in the high-resolution models. The dispersion rates are also affected by these small-scale structures as the relative horizontal dispersion coefficients are larger for the high-resolution models. The absolute vertical dispersion rates are, however, lower for the high-resolution models, because the particles tend to move along inclined eddy orbits when the resolution is low and this increases the magnitude of absolute vertical dispersion. Since relative vertical dispersion can reduce this effect from the orbital trajectories of particles, it gives a more reasonable magnitude range than absolute dispersion, and so is recommended in estimating vertical dispersion rates.

Finite size Lyapunov exponent at a saddle point

A simple stochastic system is considered modeling Lagrangian motion in a vicinity of a hyperbolic stationary point in two dimensions. We address the dependence of the Finite Size Lyapunov Exponent (FSLE) λ on the diffusivity D and the direction of the initial separation θ. It is shown that there is an insignificant difference between the curves λ=λ(θ) for pure dynamics (D=0) and for infinitely large noise (D=∞). For small D a well known boundary layer asymptotic is employed and compared with numerical results.

Defining Mediterranean and Black Sea biogeochemical subprovinces and synthetic ocean indicators using mesoscale oceanographic features.

The Mediterranean and Black Seas are semi-enclosed basins characterized by high environmental variability and growing anthropogenic pressure. This has led to an increasing need for a bioregionalization of the oceanic environment at local and regional scales that can be used for managerial applications as a geographical reference. We aim to identify biogeochemical subprovinces within this domain, and develop synthetic indices of the key oceanographic dynamics of each subprovince to quantify baselines from which to assess variability and change. To do this, we compile a data set of 101 months (2002–2010) of a variety of both “classical” (i.e., sea surface temperature, surface chlorophyll-a, and bathymetry) and “mesoscale” (i.e., eddy kinetic energy, finite-size Lyapunov exponents, and surface frontal gradients) ocean features that we use to characterize the surface ocean variability. We employ a k-means clustering algorithm to objectively define biogeochemical subprovinces based on classical features, and, for the first time, on mesoscale features, and on a combination of both classical and mesoscale features. Principal components analysis is then performed on the oceanographic variables to define integrative indices to monitor the environmental changes within each resultant subprovince at monthly resolutions. Using both the classical and mesoscale features, we find five biogeochemical subprovinces for the Mediterranean and Black Seas. Interestingly, the use of mesoscale variables contributes highly in the delineation of the open ocean. The first axis of the principal component analysis is explained primarily by classical ocean features and the second axis is explained by mesoscale features. Biogeochemical subprovinces identified by the present study can be useful within the European management framework as an objective geographical framework of the Mediterranean and Black Seas, and the synthetic ocean indicators developed here can be used to monitor variability and long-term change.

Fractional Brownian motion run with a multi-scaling clock mimics diffusion of spherical colloids in microstructural fluids.

The collective molecular reorientations within a nematic liquid crystal fluid bathing a spherical colloid cause the colloid to diffuse anomalously on a short time scale (i.e., as a non-Brownian particle). The deformations and fluctuations of long-range orientational order in the liquid crystal profoundly influence the transient diffusive regimes. Here we show that an anisotropic fractional Brownian process run with a nonlinear multiscaling clock effectively mimics this collective and transient phenomenon. This novel process has memory, Gaussian increments, and a multiscale mean square displacement that can be chosen independently from the fractal dimension of a particle trajectory. The process is capable of modeling multiscale sub-, super-, or classical diffusion. The finite-size Lyapunov exponents for this multiscaling process are defined for future analysis of related mixing processes.

Numerical investigation of chaotic advection past a groyne based on laboratory flow experiment

A laboratory flow past a groyne with complex hydrodynamics was investigated using surface Particle Tracking Velocimetry (PTV) technique for detecting chaotic features in fluvial mixing processes. In the reconstructed velocity field particles were deployed and tracked numerically in a Lagrangian way. Calculating some appropriate parameters (e.g. flushing times, finite-size Lyapunov exponent) originating from chaos theory, we are able to give a more detailed picture on surface mixing driven by aperiodic flows than traditional approaches, including the separation of sub-regions characterized by sharply different mixing efficiency.

Generalized similarity, renormalization groups, and nonlinear clocks for multiscaling.

Fixed points of the renormalization group operator Rp,rX(t)≡X(rt)/rp are said to be p-self-similar. Here X(t) is an arbitrary stochastic process. The concept of a p-self-similar process is generalized via the renormalization group operator RF,GX(t)=F[X(G(t))], where F and G are bijections on (-∞,∞) and [0,∞), respectively. If X(t) is a fixed point of RF,G, then X(t) is said to be (F,G)-self-similar. We say Y(t) is (F,G)-X(t)-similar if RF,GX(t)=Y(t) in distribution. Exit time distributions and finite-size Lyapunov exponents were obtained for these latter processes. A power law multiscaling process is defined with a multipower-law clock. This process is employed to statistically represent diffusion in a nanopore, a monolayer fluid confined between atomically structured surfaces. The tools presented provide a straightforward method to statistically represent any multiscaling process in time.

Surface water renewal and mixing mechanisms in a semi-enclosed microtidal domain. The Barcelona harbour case

Water renewal and mixing are highly related to the evolution of the ecological status in harbour areas and are crucial for environmental harbour management. For the first time, these processes have been studied in the Barcelona harbour with a high resolution 3D hydrodynamic model. This harbour is representative of a semi-enclosed domain with a complex coastline in a micro-tidal environment (like most Mediterranean harbours). The tracking of numerical Lagrangian particles deployed in the flow was used to parameterize the surface water renewal mechanisms. The use of Finite Size Lyapunov Exponents has proven to be a relatively easy and efficient way to characterize the mixing patterns. The complex geometry of the harbour and the meteo-oceanographic forcings lead to intricate hydrodynamics that define spatial heterogeneity of water renewal and mixing. The most favourable conditions for enhancing surface water renewal and mixing have been identified in idealized scenarios and realistic simulations. In general, renewal is enhanced near the harbour mouths and strongly limited in the inner areas. However, under particular forcing conditions, the residence time can decrease even in the most sheltered areas. The presence of two mouths in the harbour seems to favour water renewal in comparison to harbours with only one mouth. Mixing is mainly induced by the action of wind forcing, while external shelf currents are much less efficient. The study of two realistic simulations suggests that harbour response to real forcing cannot be deduced from the combination of simple idealized scenarios. The time evolution of the forcings and the transient response of the system play a key role in defining the residence time patterns.

Surface transport in the Northeastern Adriatic Sea from FSLE analysis of HF radar measurements

This study focuses on the surface transport in the Northeastern Adriatic Sea and the related hydrodynamic connectivity with the Gulf of Trieste (GoT) under calm or typical wind conditions: Bora (from the NE) and Sirocco (from the SE). The surface transport in the area has been investigated by evaluating the Finite-Size Lyapunov Exponents (FSLE) on the current field measured by the High Frequency (HF) coastal radar network. FSLE allow us to estimate Lagrangian Coherent Structures (LCSs), which provide information on the transport patterns associated with the flow and identify regions characterized by different dynamics. This work includes the development and set-up of the FSLE algorithm applied for the first time to the specific Adriatic area considered. The FSLE analysis during calm wind reveals an attractive LCS crossing the GoT entrance, marking the convergence between the Northern Adriatic and the outflowing gulf waters. During Bora episodes this attractive LCS is displaced westward with respect to the calm wind case, indicating that Bora drives an extended coherent outflow from the GoT. On the other hand, Sirocco wind piles up the water along the northern end of the basin. In this area an attractive LCS is found, extending mainly in the SW–NE direction. The sirocco-induced inflow of Adriatic waters in the GoT is mainly driven along its northern (Italian) side, as evidenced by the orientation of the LCS. Under Sirocco condition, as in the Bora case, there is no barrier in front of the gulf. No relevant LCSs are observed in the southern radar coverage area except for Bora cases, when a repulsive LCS develops in front of the Istrian coast separating water masses to the North and the South of it.

Do Finite-Size Lyapunov Exponents detect coherent structures?

Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here, we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.

Sea surface transport in the Western Mediterranean Sea: A Lagrangian perspective

We study the sea surface transport in the Western Mediterranean Sea from a Lagrangian point of view, in particular the Alboran and the North‐Western subbasins. The study is carried out through the analysis of 3 years of surface velocity model data through Finite Size Lyapunov Exponents, Residence Time, and virtual particle trajectories complementing the classical Eulerian approach. The spatiotemporal variability of the main transport processes is inferred from the Empirical Orthogonal Function modes of the Lyapunov Exponents, being the most relevant modes discussed and physically interpreted. Results indicate that some of the variability in the surface transport patterns in the Western Mediterranean can be explained by specific modes which provide an indication of connectivity among subbasins, like the inflow of Atlantic waters through the Ibiza Channel.

A model for quantifying oceanic transport and mesoscale variability in the Coral Triangle of the Indonesian/Philippines Archipelago

[1] The Indonesian Throughflow region (ITF) continues to pose significant research challenges with respect to its role in the global ocean circulation, the climate system, and the ecosystem sustainability in this region of maximum marine biodiversity. Complex geography and circulation features imply difficulties in both observational and numerical studies. In this work, results are presented from a newly developed high-resolution model for the Coral Triangle (CT) of the Indonesian/Philippines Archipelago specifically designed to address regional physical and ecological questions. Here, the model is used to quantify the transport through the various passages, sea surface temperature and mesoscale variability in the CT. Beyond extensive skill assessment exhibiting the model ability to represent many conspicuous features of the ITF, the high-resolution simulation is used to describe the mesoscale and submesoscale circulation through the application of Finite Size Lyapunov Exponents (FSLEs). The distribution of FSLEs is used to quantify the spatiotemporal variability in the regional mixing characteristics. The modeled seasonal and interannual variability of mixing suggests a link to large-scale climate signals such as ENSO and the Asian-Australian monsoon system.

Lagrangian transport in a microtidal coastal area: the Bay of Palma, island of Mallorca, Spain

Abstract. Coastal transport in the Bay of Palma, a small region in the island of Mallorca, Spain, is characterized in terms of Lagrangian descriptors. The data sets used for this study are the output for two months (one in autumn and one in summer) of a high resolution numerical model, ROMS (Regional Ocean Model System), forced atmospherically and with a spatial resolution of 300 m. The two months were selected because of their different wind regime, which is the main driver of the sea dynamics in this area. Finite-size Lyapunov exponents (FSLEs) were used to locate semi-persistent Lagrangian coherent structures (LCS) and to understand the different flow regimes in the bay. The different wind directions and regularity in the two months have a clear impact on the surface bay dynamics, whereas only topographic features appear clearly in the bottom structures. The fluid interchange between the bay and the open ocean was studied by computing particle trajectories and residence time (RT) maps. The escape rate of particles out of the bay is qualitatively different, with a 32% greater escape rate of particles to the ocean in October than in July, owing to the different geometric characteristics of the flow. We show that LCSs separate regions with different transport properties by displaying spatial distributions of residence times on synoptic Lagrangian maps together with the location of the LCSs. Correlations between the time-dependent behavior of FSLE and RT are also investigated, showing a negative dependence when the stirring characterized by FSLE values moves particles in the direction of escape.

Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent

In this paper, we use the finite-size Lyapunov exponent (FSLE) to characterize Lagrangian coherent structures in three-dimensional (3D) turbulent flows. Lagrangian coherent structures act as the organizers of transport in fluid flows and are crucial to understand their stirring and mixing properties. Generalized maxima (ridges) of the FSLE fields are used to locate these coherent structures. 3D FSLE fields are calculated in two phenomenologically distinct turbulent flows: a wall-bounded flow (channel flow) and a regional oceanic flow obtained by the numerical solution of the primitive equations where two-dimensional (2D) turbulence dominates. In the channel flow, autocorrelations of the FSLE field show that the structure is substantially different from the near wall to the mid-channel region and relates well to the more widely studied Eulerian coherent structure of the turbulent channel flow. The ridges of the FSLE field have complex shapes due to the 3D character of the turbulent fluctuations. In the oceanic flow, strong horizontal stirring is present and the flow regime is similar to that of 2D turbulence where the domain is populated by coherent eddies that interact strongly. This in turn results in the presence of high FSLE lines throughout the domain leading to strong non-local mixing. The ridges of the FSLE field are quasi-vertical surfaces, indicating that the horizontal dynamics dominates the flow. Indeed, due to rotation and stratification, vertical motions in the ocean are much less intense than horizontal ones. This suppression is absent in the channel flow, as the 3D character of the FSLE ridges shows.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

Random Renormalization Group Operators Applied to Stochastic Dynamics

Let X(t) be a fixed point the renormalization group operator (RGO), Rp,rX(t)=X(rt)/rp. Scaling laws for the probability density, mean first passage times, finite-size Lyapunov exponents of such fixed points are reviewed in anticipation of more general results. A generalized RGO, \(\mathcal{R}_{P,n}\) where P is a random variable, is introduced. Scaling laws associated with these random RGOs (RRGOs) are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion. The scaling laws for the RRGO are not simple power laws, but are a weighted average of power laws. The weighting used in the scaling laws can be determined adaptively via Bayes’ theorem.

MIXING IN COASTAL AREAS INFERRED FROM LYAPUNOV EXPONENTS: IMPLICATIONS FOR TRANSPORT

We study the horizontal surface mixing and the transport induced by waves, using local Lyapunov exponents and high resolution data from numerical simulations of waves and currents. By choosing the proper spatial (temporal) parameters we compute the Finite Size and Finite Time Lyapunov exponents (FSLE and FTLE) focussing on the local stirring and diffusion inferred from the Lagrangian Coherent Structures (LCS). The methodology is tested by deploying a set of eight lagrangian drifters and studying the path followed against LCS derived under current field and waves and currents.

Seasonal and regional characterization of horizontal stirring in the global ocean

Recent work on Lagrangian descriptors has shown that Lyapunov Exponents can be applied to observed or simulated data to characterize the horizontal stirring and transport properties of the oceanic flow. However, a more detailed analysis of regional dependence and seasonal variability was still lacking. In this paper, we analyze the near‐surface velocity field obtained from the Ocean general circulation model For the Earth Simulator (OFES) using Finite‐Size Lyapunov Exponents (FSLE). We have characterized regional and seasonal variability. Our results show that horizontal stirring, as measured by FSLEs, is seasonally‐varying, with maximum values in Summer time. FSLEs also strongly vary depending on the region: we have first characterized the stirring properties of Northern and Southern Hemispheres, then the main oceanic basins and currents. We have finally studied the relation between averages of FSLE and some Eulerian descriptors such as Eddy Kinetic Energy (EKE) and vorticity (ω) over the different regions.