No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
A Comparison of Finite-Time and Finite-Size Lyapunov Exponents
Abstract
Finite-time and finite-size Lyapunov exponents are related concepts that have been used for the purpose of identifying transport structures in time-dependent flow. The preference for one or the other concept seems to be based more on a tradition within a scientific community than on proven advantages. In this study, we demonstrate that with the two concepts highly similar visualizations can be produced, by maximizing a simple similarity measure. Furthermore, we show that results depend crucially on the numerical implementation of the two concepts.
... The second main contribution of this work is to provide a theoretical link between the FTLE and the ISLE. Even though these two quantities are measuring different properties of the flow, it has been reported widely, such as [26], that they give visually very similar solutions in many examples. We are going to show in particular that their ridges can be identified through the ridge of the largest eigenvalue of the associated Cauchy-Green deformation tensor of the flow. ...
... [4] has pointed out that the FTLE might be unable to recognize the boundaries between the chaotic and the large-scale mixing regime. [16] has argued that the FSLE has several limitations in Lagrangian coherence detection including aspects from local ill-posedness, spurious ridges and intrinsic jump-discontinuities. Nevertheless, [26] has stated that these two concepts might yield similar results, if properly calibrated, and could be even interchangeable in some applications. Here, we would like to emphasize that the main purpose of this work is on the computational aspect of using these Lyapunov exponents, rather than on comparing the advantages and the disadvantages of these different methods. ...
... They are two different tools to study how rapid λ t 0 (x) grows. A careful comparison of these two quantities can be found in, for example, [26]. ...
We propose efficient Eulerian numerical approaches for constructing flow maps in continuous dynamical systems and apply these methods to compute the finite time Lyapunov exponent (FTLE), the finite size Lyapunov exponent (FSLE) and also a related infinitesimal size Lyapunov exponent (ISLE). The new algorithm improves the original formulation developed in our previous works so that the associated partial differential equations (PDEs) are solved forward in time and, therefore, the forward flow map can now be determined on the fly. When applied to the ISLE computations, the Eulerian method is computationally efficient. For each separation factor $r$ in the definition of the FSLE or the ISLE, typical Lagrangian methods require to shoot and monitor an individual set of ray trajectories. If the scale $r$ is changed, these methods have to restart the whole computations all over again. The proposed method, however, requires to extract only an isosurface of a volumetric data for an individual value of $r$ which can be easily done using any well-developed efficient interpolation method or simply an isosurface extraction algorithm. Moreover, we provide a theoretical link between the FTLE and the ISLE fields which explains the similarity in these solutions observed in various applications.
... On the contrary, FTLEs have been shown to better capture recirculation regions than FSLEs (Sadlo and Peikert, 2007). In a recent paper, Peikert et al. (2014) show that, if properly calibrated by similarity measures, both FTLEs and FSLEs may produce comparable results that can be interchangeably used for most purposes in flow visualizations. Further investigation, especially in the context of realistic geophysical flows, will thus provide valuable information on the mutual importance of the Lagrangian measures, namely FTLE and FSLE. ...
... Following Peikert et al. (2014), we compare FTLE and FSLE maps by calculating their correlation coefficient. FTLE and FSLE fields adopted for the analysis are obtained by seeding of an initial grid with a regular spacing of 200 m. ...
The present work aims to detect Lagrangian transport barriers in the Gulf of Trieste by means of Lyapunov-exponent approach and tensorlines of the Cauchy-Green tensor. Lagrangian Coherent Structures (LCSs) are calculated employing 2D surface velocity fields measured by the coastal radars of the TOSCA EU research project (Tracking Oil Spills & Coastal Awareness Network). Moreover, surface drifters were deployed during the project. Comparisons between Eulerian velocity of HF-radar fields and Lagrangian velocity of drifters are carried out alongside single-particle tracking reliability. In particular, the possible influence of the data gaps in the HF-radar fields have been carefully considered. LCSs have proven to be robust against the quality of the starting HF-radar fields, leading to helpful insights in drifter positions. Indeed, after 24-hour integration the observed position of the drifter is approximately 1.5 km far from the nearest LCS, while a standard approach based on single-particle computations leads to larger errors (up to 5–7 km). However, such result must be properly interpreted taking into account the elongated nature of LCSs. A comparison between two common diagnostic tools of Lagrangian barriers is performed: Finite-Time and Finite-Size Lyapunov Exponent fields are compared in order to assess whether the patterns detected by the two measures are comparable. Finally, a joint analysis between LCSs and single-particle tracking is carried out and the results suggest that it would be desirable to couple these two approaches in real applications.
... In fact, the sought-after seed curves could be seen as loci of points where the corresponding dynamical system (represented by the input vector field) is as unstable as possible. This is formalized by the notion of Lyapunov exponents [25,33]. In our approach, we employ the local variant of the Lyapunov exponent [6]. ...
... Our seed curves are integral curves that correspond to the eigenvectors associated to the largest local Lyapunov exponents. As a promising avenue for future research, it would be interesting to look at finite-time Lyapunov exponents [25,15]. However, one would need to also investigate the influence of the selected time-interval on the obtained results, something we expect to be considerably more expensive than our less global method given the complexity of the computation of the non-local Lyapunov exponents. ...
We investigate a class of stream surfaces that expand in time as much as possible. Given a vector field, we look for seed curves that locally propagate in time in a stretch-maximizing manner, i.e., curves that infinitesimally expand most progressively. We show that such a curve is generically unique at every point in an incompressible flow and offers a very good initial guess for a stretch-maximizing stream surface. With the application of efficient fluid advection-diffusion in mind, we optimize fluid injection towards optimal advection and show several examples on benchmark datasets.
... Ref. [26] conducted an assessment of both methods and pointed out advantages that FTLE has over FSLE. Nonetheless, Ref. [27] highlighted in their comparison that, with proper calibration, FTLE and FSLE can lead to similar results. In this work, we extend prior work on 2D FTLE [9] to a generalized 3D representation. ...
In this work, we introduce a scalable and efficient GPU-accelerated methodology for volumetric particle advection and finite-time Lyapunov exponent (FTLE) calculation, focusing on the analysis of Lagrangian coherent structures (LCS) in large-scale direct numerical simulation (DNS) datasets across incompressible, supersonic, and hypersonic flow regimes. LCS play a significant role in turbulent boundary layer analysis, and our proposed methodology offers valuable insights into their behavior in various flow conditions. Our novel owning-cell locator method enables efficient constant-time cell search, and the algorithm draws inspiration from classical search algorithms and modern multi-level approaches in numerical linear algebra. The proposed method is implemented for both multi-core CPUs and Nvidia GPUs, demonstrating strong scaling up to 32,768 CPU cores and up to 62 Nvidia V100 GPUs. By decoupling particle advection from other problems, we achieve modularity and extensibility, resulting in consistent parallel efficiency across different architectures. Our methodology was applied to calculate and visualize the FTLE on four turbulent boundary layers at different Reynolds and Mach numbers, revealing that coherent structures grow more isotropic proportional to the Mach number, and their inclination angle varies along the streamwise direction. We also observed increased anisotropy and FTLE organization at lower Reynolds numbers, with structures retaining coherency along both spanwise and streamwise directions. Additionally, we demonstrated the impact of lower temporal frequency sampling by upscaling with an efficient linear upsampler, preserving general trends with only 10% of the required storage. In summary, we present a particle search scheme for particle advection workloads in the context of visualizing LCS via FTLE that exhibits strong scaling performance and efficiency at scale. Our proposed algorithm is applicable across various domains, requiring efficient search algorithms in large, structured domains. While this article focuses on the methodology and its application to LCS, an in-depth study of the physics and compressibility effects in LCS candidates will be explored in a future publication.
... Surface current vectors alone may not adequately identify submesoscale features, such as fronts and eddies, due to the time-dependent nature of processes in the marine environment [67,68]. Instead, to characterize the strength and persistence of these features, we use a Lagrangian (i.e. ...
Marine predators face the challenge of reliably finding prey that is patchily distributed in space and time. Predators make movement decisions at multiple spatial and temporal scales, yet we have a limited understanding of how habitat selection at multiple scales translates into foraging performance. In the ocean, there is mounting evidence that submesoscale (i.e. less than 100 km) processes drive the formation of dense prey patches that should hypothetically provide feeding hot spots and increase predator foraging success. Here, we integrated environmental remote-sensing with high-resolution animal-borne biologging data to evaluate submesoscale surface current features in relation to the habitat selection and foraging performance of blue whales in the California Current System. Our study revealed a consistent functional relationship in which blue whales disproportionately foraged within dynamic aggregative submesoscale features at both the regional and feeding site scales across seasons, regions and years. Moreover, we found that blue whale feeding rates increased in areas with stronger aggregative features, suggesting that these features indicate areas of higher prey density. The use of fine-scale, dynamic features by foraging blue whales underscores the need to take these features into account when designating critical habitat and may help inform strategies to mitigate the impacts of human activities for the species.
... Karrasch and Haller [36] made an assesment of both methods and pointed out advantages that FTLE has over FSLE. However, in their comparison, Peiker et al. [37] concluded that, if properly calibrated, FTLE and FSLE can produce comparable results. A two-dimensional FTLE approach has been selected for this study. ...
View Video Presentation: https://doi.org/10.2514/6.2022-3627.vid Lagrangian Coherent Structures (LCS) have recently received a significant attention due to its advantages over Eulerian coherent structure identification schemes. Transport barriers and transport enhancers as material surfaces are identified by LCS techniques and can be used to analyze turbulent mixing in many engineering applications. This study utilized high-fidelity Direct Numerical Simulation (DNS) databases of spatially-developing turbulent boundary layers (SDTBL) at the incompressible, supersonic (Mach = 2.5), and hypersonic (Mach = 5) flow regimes. The main purpose is to qualitatively study the effects of flow compressibility and Reynolds number on LCS. Compressibility effects on turbulent coherent structures were observed to be weak, becoming more noticeable in the hypersonic regime for low Reynolds cases. Furthermore, the presence of hairpin vortices was scarce beyond y+ = 100 (i.e., in the log-wake region) of hypersonic turbulent boundary layers. In contrast, the Reynolds number dependency on LCS was evident given by the high level of isotropization of turbulent coherent structures. Moreover, strong compressibility effects were observed once the Reynolds number was increased. This resulted in more abundant and more isotropic structures for high Reynolds supersonic flow.
... The basic idea of this method consists in measuring the time needed to achieve a given separation from each location, thereby substituting a size parameter to the time parameter found in the definition of FTLE. Direct comparisons between FTLE and FSLE have been made by Boffetta et al. [5], Sadlo and Peikert [33], and Peikert et al. [27]. Most recently, Karrasch and Haller showed that FSLE was prone to inaccuracies and false positives in the characterization of LCS [19]. ...
The computation of Lagrangian coherent structures (LCS) has established itself as a prominent means to reveal significant geometric structures in time-dependent vector fields. Their characterization, however, requires the selection of a suitable time parameter for the construction of the flow map that may not be known in advance. We present in this paper a continuous time-scale framework for LCS extraction and visualization. Specifically, we treat the time axis as a continuum from which a best temporal scale is automatically determined at each spatial location for the extraction of LCS. Beyond its effectiveness with vector fields we show that this method can be successfully applied to improve the characterization of salient structures in tensor fields and discrete maps. We present applications of our method to problems spanning fluid dynamics, medical imaging, and orbital mechanics. The results show that our approach can reveal important structural features that are missed by existing LCS extraction methods.
... At every hourly time step, the trajectories of the evenly spaced grid of tracers were integrated for the preceding 6, 12, 24 or 96 h period and FTLE was calculated from the time-dependent movement of tracer trajectories. Similar to finitesize Lyapunov exponents (FSLE) (Peikert et al., 2014), FTLE calculates areas of potential particle accumulation e including zooplankton e based on the movement of surface currents and has been show to correlate with foraging animal movement patterns (Abrahms et al., 2018;Scales et al., 2014;Scales et al., 2017). A time-series visualization of the FTLE values in the study region was created to assess both the size and temporal persistence of Lagrangian coherent structures (LCS), which were visually identified ridges of high FTLE values (Shadden et al., 2005; Video S4). ...
Large groups of animals aggregate around resource hotspots, with group size often influenced by the heterogeneity of the environment. In most cases, the foraging success of individuals within groups is interdependent, scaling either constructively or destructively with group size. Here we used biologging tags, acoustic prey mapping, passive acoustic recording of social cues and remote sensing of surface currents to investigate an alternative scenario in which large, dense aggregations of southeast Atlantic humpback whales, Megaptera novaeangliae, and northeast Pacific blue whales, Balaenoptera musculus, were each associated with ephemeral krill aggregations large enough such that their availability to predators appeared to be influenced more by environmental features than by consumption, implying independence of group size and consumption rates. We found that the temporal scale and spatial extent of oceanographic drivers were consistent with the temporal scale and locations of predator aggregations, and additionally found that groups formed above bathymetric features known to promote zooplankton concentration. Additionally, we found calling behaviour counter-indicative of competition: blue whale foraging calls were anomalously high during observed aggregation time periods, suggesting signalling behaviour that could alert conspecifics to the location of high-quality resources. Modelled results suggest that the use of social information reduces the time required for individuals to discover and exploit high-quality resources, allowing for more efficient foraging without apparent costs to the caller. Thus, rorqual whales foraging in these environments appear to exhibit a social foraging strategy whereby a behaviour with negligible individual costs (signalling) provides information that enhances group foraging efficiency. The population density dependence of this social foraging strategy may help explain why some rorqual species were at first slow to recover from human exploitation, but have since increased more rapidly.
... The locations of high values of backward and forward FTLE signify the areas for strongest clustering and spreading potentials, respectively. Because of the difference in the dynamics between consecutive temporal windows, the FTLE fields are normalised by the basin-wide maximum FTLE value as recommended in (Peikert et al., 2014). Therefore, a value close to one signifies areas of strong activity of clustering or spreading. ...
We explore the possibility to identify areas of intense patch formation from floating items due to systematic convergence of surface velocity fields by means of a visual comparison of Lagrangian Coherent Structures (LCS) and estimates of areas prone to patch formation using the concept of Finite-Time Compressibility (FTC, a generalisation of the notion of time series of divergence). The LCSs are evaluated using the Finite Time Lyapunov Exponent (FTLE) method. The test area is the Gulf of Finland (GoF) in the Baltic Sea. A basin-wide spatial average of backward FTLE is calculated for the GoF for the first time. This measure of the mixing strength displays a clear seasonal pattern. The evaluated backward FTLE features are linked with potential patch formation regions with high FTC levels. It is shown that areas hosting frequent upwelling or downwelling have consistently stronger than average mixing intensity. The combination of both methods, FTC and LCS, has the potential of being a powerful tool to identify the formation of patches of pollution at the sea surface.
... In this analysis, the finite time Lyapunov exponent (FTLE) is employed as a diagnostic quantity for the LCSs. This diagnostic approach among others is known for the simplicity and objectivity of its algorithms and serving as proxies for LCSs [Peikert et al., 2014;Hadjighasem et al., 2017]. Comparison across different detection methods is presented in detail in Hadjighasem et al. [2017]. ...
Coastal tidal estuaries are vital to the exchange of energy and material between inland waters and the open ocean. Debris originating from the land and ocean enter this environment and are transported by flow current (river outflow and tide), wind and waves. Understanding and predicting the source and fate of such debris has considerable environmental, economic and visual importance. We show that this issue can be addressed using the Lagrangian coherent structures (LCS) technique which is highly robust to hydrodynamic model uncertainties. Here we present a comprehensive study showing the utility of this approach to describe the fate of floating material in a coastal tidal embayment. An example is given from Moreton Bay, a semi-enclosed subtropical embayment with high morphologic, ecological and economic significance to Southeast Queensland, Australia. Transport barriers visualised by the LCS create pathways and barriers for material transport in the embayment. It was found that the wind field modified both the strength and location of the transport barriers. One of the key outcomes is the demonstration of the significant role of islands in partitioning the transport of material and mixing within the embayment. The distribution of the debris sources along the shoreline are explained by the relative location of the LCS to the shoreline. Therefore, extraction of LCS can help to predict sources and fate of anthropogenic marine debris and thus, serve as a useful way for effective management of vulnerable regions and marine protected areas.
... Vortex-like structures, so-called rotation cells, are visible in both, in-and out-degree, "north" and "south" of the meander with an opposite direction of rotation. More detailed numerical analysis (not shown) reveals that the resulting structures of in-and out-degrees closely match those of the timebackward and time-forward finite-time Lyapunov exponents [68], respectively, as expected from previous studies [40,10]. Specifically, since ridges in the FTLE field have been shown to be transport barriers [69], we expect the same to hold for ridges in the degree field. ...
During the last years, complex network approaches have demonstrated their great potentials as versatile tools for exploring the structural as well as dynamical properties of complex systems from a variety of different fields. Among others, recent successful examples include their application to studying flow systems in both, abstract mathematical and real-world geophysical contexts. In this context, two recent developments are particularly notable: on the one hand, correlation-based functional network approaches allow inferring statistical interrelationships, for example between macroscopic regions of the Earth’s climate system, which are hidden to more classical statistical analysis techniques. On the other hand, Lagrangian flow networks provide a new tool to identify dynamically relevant structures in atmosphere, ocean or, more generally, the phase space of complex systems. This chapter summarizes these recent developments and provides some illustrative examples highlighting the application of both concepts to selected paradigmatic low-dimensional model systems.
... Bettencourt et al. [7] relied on a quasi-2D approximation neglecting the vertical separations of the trajectories after evolving them in the 3D velocity field. (They computed finite-size Lyapunov exponents (FSLE) instead of FTLE, but the metrics generally produce similar results [42] .) Moreover, the GCMs used in those studies solved for the vertical velocity diagnostically, as is typical for regional ocean models. ...
The linearized 3D Euler equations on an f-plane with constant stratification admit a family of analytical wave solutions. Here, we investigate the Lagrangian properties of one such solution, a standing wave quadrupole, whose simplicity and symmetry make it an ideal time-varying 3D testbed for developing dynamical systems methods. In spite of its simplicity, the Eulerian solution gives rise to highly complex transport structures. Particle trajectories wind around tori-like surfaces with varying cross-sections. They are generally governed by the internal wave frequency plus subinertial frequencies, which depend on starting locations. The spatial variation in this subinertial period produces mixing in the periodic wave motion, a process completely distinct from diapycnal mixing typically associated with internal waves. Nonetheless, finite-time Lyapunov exponents, calculated from the 3D velocity field, clearly delineate transport barriers. These barriers identify five types of coherent Lagrangian structures, which oscillate at the super-inertial internal wave frequency. Two of these types are solely located near the surface, extending to depths unassociated with any Eulerian flow characteristic. The discovery of such shallow structures in the absence of a related Eulerian signal raises the interesting question whether similar structures may be hiding in the real ocean.
... However, it has been shown that, although related, the FSLE can be somewhat different from FTLE, particularly depending on details such as the numerical implementation and the particle seeding. 44 ...
Complex flows mix efficiently, and this process can be understood by considering the stretching and folding of material volumes. Although many metrics have been devised to characterize stretching, fewer are able to capture folding in a quantitative way in spatiotemporally variable flows. Here, we extend our previous methods based on the finite-time curving of fluid-element trajectories to nonzero scales and show that this finite-scale finite-time curvature contains information about both stretching and folding. We compare this metric to the more commonly used finite-time Lyapunov exponent and illustrate our methods using experimental flow-field data from a quasi-two-dimensional laboratory flow. Our new analysis tools add to the growing set of Lagrangian methods for characterizing mixing in complex, aperiodic fluid flows.
... Waugh et al. [7] also examined the relationship between finite-time Lyapunov exponents and relative dispersion for a simple uniform strain flow and ocean flow. Peikert et al. [8] developed a comparison between Finite-Time and Finite-Size Lyapunov Exponents to analyze which one of these methods better describes the properties of flows. ...
In the present work we implement and
employ Lagrangian transport models for basin scale
circulation, with a particular focus on the Adriatic Sea in
order to analyze and estimate the Finite-Time Lyapunov
Exponent (FTLE) fields, the Mean Kinetic Energy and the
Eddy Kinetic Energy of the flow. We compare FTLE fields
obtained for two different years as sample cases (2007 with
mild Winter and cold Autumn, 2008 with normal Winter
and hot Summer) to study the superficial flow properties.
Results obtained for FTLE maps and MKE show the
effects on general circulation triggered by wind
forcing and by the Po River outflow, on the value of transit
and residence time of numerical particles in the Adriatic
Sea.
... See [6,9,3], to name a few. For the nonautonomous case, Lagrangian Coherent Structures (LCS) based on Finite Time Lyapunov Exponents (FTLE) that focus on maximal local stretching has become a popular computational way to study transport, see [12,13], but with caveats regarding the possibility of false positives have been revealed [23]; similarly Finite Sized Lyapunov Exponents (FSLE) show differences and likely false positives, [29]. On the other hand, transfer operator methods based on Galerkin-Ulam matrices which evolve distributions of ensembles of initial conditions have also been successfully developed for finding so-defined coherent pairs, [8,9,11]. ...
Mixing, and coherence are fundamental issues at the heart of understanding
transport in fluid dynamics and other non-autonomous dynamical systems.
Recently, the notion of coherence has come to a more rigorous footing, and
particularly within the recent advances of finite-time studies of
non-autonomous dynamical systems. Here we define shape coherent sets as a means
to emphasize the intuitive notion of ensembles which "hold together" for some
period of time, and we contrast this notion to other recent perspectives of
coherence, notably "coherent pairs", and likewise also to the geodesic theory
of material lines. We will relate shape coherence to the differential geometry
concept of curve congruence through matching curvatures. We show that points in
phase space where there is a zero-splitting between stable and unstable
manifolds locally correspond to points where curvature will evolve only slowly
in time. Then we develop curves of points with zero-angle, meaning
non-hyperbolic splitting, by continuation methods in terms of the implicit
function theorem. From this follows a simple ODE description of the boundaries
of shape coherent sets. We will illustrate our methods with popular benchmark
examples, and further investigate the intricate structure of foliations
geometry.
... [22, Sec. 10.5.1] and [27]. ...
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.
In the marine environment, dynamic physical processes shape biological productivity and predator–prey interactions across multiple scales. Identifying pathways of physical–biological coupling is fundamental to understand the functioning of marine ecosystems yet it is challenging because the interactions are difficult to measure. We examined submesoscale (less than 100 km) surface current features using remote sensing techniques alongside ship-based surveys of krill and baleen whale distributions in the California Current System. We found that aggregative surface current features, represented by Lagrangian coherent structures (LCS) integrated over temporal scales between 2 and 10 days, were associated with increased (a) krill density (up to 2.6 times more dense), (b) baleen whale presence (up to 8.3 times more likely) and (c) subsurface seawater density (at depths up to 10 m). The link between physical oceanography, krill density and krill–predator distributions suggests that LCS are important features that drive the flux of energy and nutrients across trophic levels. Our results may help inform dynamic management strategies aimed at reducing large whales ship strikes and help assess the potential impacts of environmental change on this critical ecosystem.
In this work, we introduce a scalable and efficient GPU-accelerated methodology for volumetric particle advection and finite-time Lyapunov exponent (FTLE) calculation, focusing on the analysis of Lagrangian Coherent Structures (LCS) in large-scale Direct Numerical Simulation (DNS) datasets across incompressible, supersonic, and hypersonic flow regimes. LCS play a significant role in turbulent boundary layer analysis, and our proposed methodology offers valuable insights into their behavior in various flow conditions. Our novel owning-cell locator method enables efficient, constant-time cell search, and the algorithm draws inspiration from classical search algorithms and modern multi-level approaches in numerical linear algebra. The proposed method is implemented for both multi-core CPUs and Nvidia GPUs, demonstrating strong scaling up to 32,768 CPU cores and up to 62 Nvidia V100 GPUs. By decoupling particle advection from other problems, we achieve modularity and extensibility, resulting in consistent parallel efficiency across different architectures. Our methodology was applied to calculate and visualize the FTLE on four turbulent boundary layers at different Reynolds and Mach numbers, revealing that coherent structures grow more isotropic proportional to the Mach number, and their inclination angle varies along the streamwise direction. We also observed increased anisotropy and FTLE organization at lower Reynolds numbers, with structures retaining coherency along both spanwise and streamwise directions. Additionally, we demonstrated the impact of lower temporal frequency sampling by upscaling with an efficient linear upsampler, preserving general trends with only 10% of the required storage. In summary, we present a particle search scheme for particle advection workloads in the context of visualizing LCS via FTLE that exhibits strong scaling performance and efficiency at scale. Our proposed algorithm is applicable across various domains requiring efficient search algorithms in large structured domains. While this manuscript focuses on the methodology and its application to LCS, an in-depth study of the physics and compressibility effects in LCS candidates will be explored in a future publication.
While computer simulations typically store data at the highest available spatial resolution, it is often infeasible to do so for the temporal dimension. Instead, the common practice is to store data at regular intervals, the frequency of which is strictly limited by the available storage and I/O bandwidth. However, this manner of temporal subsampling can cause significant errors in subsequent analysis steps. More importantly, since the intermediate data is lost, there is no direct way of measuring this error after the fact. One particularly important use case that is affected is the analysis of unsteady flows using pathlines, as it depends on an accurate interpolation across time. Although the potential problem with temporal undersampling is widely acknowledged, there currently does not exist a practical way to estimate the potential impact. This paper presents a simple-to-implement yet powerful technique to estimate the error in pathlines due to temporal subsampling. Given an unsteady flow, we compute pathlines at the given temporal resolution as well as subsamples thereof. We then compute the error induced due to various levels of subsampling and use it to estimate the error between the given resolution and the unknown ground truth. Using two turbulent flows, we demonstrate that our approach, for the first time, provides an accurate, a posteriori error estimate for pathline computations. This estimate will enable scientists to better understand the uncertainties involved in pathline-based analysis techniques and can lead to new uncertainty visualization approaches using the predicted errors.
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.
We explore the possibility to identify areas of intense patch formation from floating items due to systematic convergence of surface velocity fields by means of a visual comparison of Lagrangian Coherent Structures (LCS) and estimates of areas prone to patch formation using the concept of Finite-Time Compressibility (FTC, a generalisation of the notion of time series of divergence). The LCSs are evaluated using the Finite Time Lyapunov Exponent (FTLE) method. The test area is the Gulf of Finland in the Baltic Sea. A basin-wide spatial average of backward FTLE is calculated for the GoF for the first time. This measure of the mixing strength displays a clear seasonal pattern. The evaluated backward FTLE features are linked with potential patch formation regions with high FTC levels. It is shown that areas hosting frequent upwelling or downwelling have consistently stronger than average mixing intensity. The combination of both methods, FTC and LCS, has the potential of being a powerful tool to identify the formation of patches of pollution at the sea surface.
Coastal tidal estuaries are vital to the exchange of energy and material between inland waters and the open ocean. Debris originating from the land and ocean enter this environment and are transported by currents (river outflow and tide), wind, waves and density gradients. Understanding and predicting the source and fate of such debris has considerable environmental, economic and visual importance. We show that this issue can be addressed using the Lagrangian coherent structures (LCS) technique which is highly robust to hydrodynamic model uncertainties. Here we present a comprehensive study showing the utility of this approach to describe the fate of floating material in a coastal tidal embayment. An example is given from Moreton Bay, a semi-enclosed subtropical embayment with high morphologic, ecological and economic significance to Southeast Queensland, Australia. Transport barriers visualised by the LCS create pathways and barriers for material transport in the embayment. It was found that the wind field modified both the rate attraction and location of the transport barriers. One of the key outcomes is the demonstration of the significant role of islands in partitioning the transport of material and mixing within the embayment. The distribution of the debris sources along the shoreline are explained by the relative location of the LCS to the shoreline. Therefore, extraction of LCS can help to predict sources and fate of anthropogenic marine debris and thus, serve as a useful way for effective management of vulnerable regions and marine protected areas.
In the last decades many possible applications of nonlinear dynamics in communication systems and signal processing have been reported. Conversely, techniques usually employed by the signal processing and communication systems communities, as correlation, power spectral density analysis, and linear filters, among others have been used to characterize chaotic dynamical systems. This chapter presents four works that aim to use tools from both fields to generate new and interesting results: (1) a message authentication system based on chaotic fingerprint; (2) a study of the spectral characteristics of the chaotic orbits of the Hénon map; (3) an investigation of the chaotic nature of the signals generated by a filtered Hènon map, and (4) a communication system that presents equalization and a switching scheme between chaos-based and conventional modulations.
Flow is one of the most fundamental phenomenon in natural world. Flow visualization can give an intuitive view of flow fields and contribute to the observation and analysis of flow. Real flow fields are all dynamic while traditional visualization approaches are not able to reflect the trends of flows or the diversity inside them. Based on the research of Lagrangian Coherent Structures in Computational Fluid Dynamics, the concept of flow field topology was redefined, two typical 2D dynamic vector fields were selected for topology analysis, and the results were used to optimize the visual effects of geometric visualization. Experiments show that visualization of 2D dynamic vector fields based on topology analysis can extract the coherent structures of flows which are helpful to the comprehension of flow for researchers and description of characteristic difference inside flows. © 2016, The Editorial Board of Journal of System Simulation. All right reserved.
Blood flow data from direct measurements (4D flow MRI) or numerical simulations opens new possibilities for the understanding of the development of cardiac diseases. However, before this new data can be used in clinical studies or for diagnosis, it is important to develop a notion of the characteristics of typical flow structures. To support this process we developed a novel blood flow clustering and exploration method. The method builds on the concept of coherent flow structures. Coherence maps for cross-sectional slices are defined to show the overall degree of coherence of the flow. In coherent regions the method summarizes the dominant blood flow using a small number of pathline representatives. In contrast to other clustering approaches the clustering is restricted to coherent regions and pathlines with low coherence values, which are not suitable for clustering and thus are not forced into clusters. The coherence map is based on the Finite-time Lyapunov Exponent (FTLE). It is created on selected planes in the inflow respective outflow area of a region of interest. The FTLE value measures the rate of separation of pathlines originating from this plane. Different to previous work using FTLE we do not focus on separating extremal lines but on local minima and regions of low FTLE intensities to extract coherent flow. The coherence map and the extracted clusters serve as basis for the flow exploration. The extracted clusters can be selected and inspected individually. Their flow rate and coherence provide a measure for their significance. Switching off clusters reduces the amount of occlusion and reveals the remaining part of the flow. The non-coherent regions can also be explored by interactive manual pathline seeding in the coherence map.
Typical fluid particle trajectories are sensitive to changes in their initial conditions. This makes the assessment of flow models and observations from individual tracer samples unreliable. Behind complex and sensitive tracer patterns, however, there exists a robust skeleton of material surfaces, Lagrangian coherent structures (LCSs), shaping those patterns. Free from the uncertainties of single trajectories, LCSs frame, quantify, and even forecast key aspects of material transport. Several diagnostic quantities have been proposed to visualize LCSs. More recent mathematical approaches identify LCSs precisely through their impact on fluid deformation. This review focuses on the latter developments, illustrating their applications to geophysical fluid dynamics.
This is a minimum mathematical introduction to Donating Regions, a concept introduced in [Q. Zhang, SIAM Review, 55(3), 2013] and followed up in a number of subsequent papers.
While most large-scale studies of population connectivity use numerical ocean model output to compute statistics of thousands of virtual particles, large uncertainties persist as models have limited spatial resolution and often do not reproduce observed ocean variability. Furthermore, scale-dependent dispersion on the surface ocean remains an important open subject in physical oceanography. Lagrangian observations from surface drifters come with their own set of problems, most notably limited numbers, sampling bias, finite lifetime, and position uncertainties from wind and wave effects. Despite their limitations, surface drifter trajectories provide important observations of surface ocean transport.
Twenty six GPS-tracked surface drifters were deployed in the central Adriatic Sea in May 2013 to investigate surface transport and to identify Lagrangrian pathways. Our particular interest are transit times between the Italian peninsula and the east coast (Croatia and Albania) as well as transit times between the network of marine protected areas (MPAs) in the Adriatic Sea. Preliminary results reveal transit times between MPAs in the central and southern Adriatic that vary from a few days to a few months. None of the drifters released in this experiment reached the Albanian coast. The drifters exited the Adriatic Sea by the end of July 2013 with several traveling to the Ionian Sea.
The drifter trajectories observed were determined by ambient environmental conditions (currents, winds, waves, stratification, etc.). Therefore, to determine the degree to which the present results can be extrapolated, the May 2013 drifter trajectories will be compared to a historical dataset of over 300 drifter trajectories in the Adriatic Sea. Additionally, the atmospheric and oceanographic conditions during the May 2013 experiment will be compared to climatological conditions. Future work will include tracking of thousands of virtual particles using output from a Regional Ocean Modeling System (ROMS) simulation of the Adriatic Sea during the same time period.
Lagrangian coherent structures are time-evolving surfaces that highlight areas in flow fields where neighboring advected particles diverge or converge. The detection and understanding of such structures is an important part of many applications such as in oceanography where there is a need to predict the dispersion of oil and other materials in the ocean. One of the most widely used tools for revealing Lagrangian coherent structures has been to calculate the finite-time Lyapunov exponents, whose maximal values appear as ridgelines to reveal Lagrangian coherent structures. In this paper we explore an alternative formulation of Lyapunov exponents for computing Lagrangian coherent structures.
We use recent developments in the theory of finite-time dynamical systems to
objectively locate the material boundaries of coherent vortices in
two-dimensional Navier--Stokes turbulence. We show that these boundaries are
optimal in the sense that any closed curve in their exterior will lose
coherence under material advection. Through a detailed comparison, we find that
other available Eulerian and Lagrangian techniques significantly underestimate
the size of each coherent vortex.
Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finite-size Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the underlying hyperbolic structure of the flow. For the first time, the exchange mechanism associated with lobe dynamics is characterized using atmospheric analyzed winds. The tangling manifolds form a stochastic layer around the vortex. It is found that fluid is not only expelled from this layer towards the surf zone, but also is injected inward from the surf zone, through a process similar to the turnstile mechanism in lobe dynamics. The vortex edge, defined as the location of the maximum gradient in potential vorticity or tracer, is found to be the southward (poleward) envelope of this stochastic layer. Exchanges with the inside of the vortex are therefore largely decoupled from those, possibly intense exchanges, between the stochastic layer and the surf zone. We stress that using the kinematic boundary defined by the hyperbolic points and the manifolds as an operational definition of vortex boundary is not only unpractical but also leads to spurious estimates of exchanges. We also present preliminary results of diagnostics based on rigorous theory of finite-time hyperbolicity.
▪ Abstract Chaotic advection and, more generally, ideas from dynamical systems, have been fruitfully applied to a diverse, and varied, collection of mixing and transport problems arising in engineering applications over the past 20 years. Indeed, the “dynamical systems approach” was developed, and tested, to the point where it can now be considered a standard tool for understanding mixing and transport issues in many disciplines. This success for engineering-type flows motivated an effort to apply this approach to transport and mixing problems in geophysical flows. However, there are fundamental difficulties arising in this endeavor that must be properly understood and overcome. Central to this approach is that the starting point for analysis is a velocity field (i.e., the “dynamical system”). In many engineering applications this can be obtained sufficiently accurately, either analytically or computationally, so that it describes particle trajectories for the actual flow. However, in geophysical flows (a...
The popularity of vector field topology in the visualization community is due mainly to the topological skeleton which captures the essential information on a vector field in a set of lines or surfaces separating regions of different flow behavior. Unfortunately, vector field topology has no straightforward extension to unsteady flow, and the concept probably most closely related to the topological skeleton are the so-called Lagrangian coherent structures (LCS). LCS are material lines or material surfaces that separate regions of different flow behavior. Ideally, such structures are material lines (or surfaces) in an exact sense and at the same time maximally attracting or repelling, but practical realizations such as height ridges of the finite-time Lyapunov exponent (FTLE) fulfill these two requirements only in an approximate sense. In this paper, we quantify the deviation from exact material lines/surfaces for several FTLE-based concepts, and we propose a numerically simpler variants of FTLE ridges that has equal or better error characteristics than classical FTLE height ridges.
The finite-time Lyapunov exponent (FTLE) is useful for the visualization of time-dependent velocity fields. The ridges of this derived scalar field have been shown to correspond well to attracting or repelling material structures, so-called Lagrangian coherent structures (LCS). There are two issues involved in the computation of FTLE for this purpose. Firstly, it is often not practically possible to refine the grid for sampling the flow map until convergence of FTLE is reached. Slow conversion is mostly caused by gradient underestimation. Secondly, there is a parameter, the integration time, which has to be chosen sensibly. Both of these problems call for an examination in scale-space. We show that a scale-space approach solves the problem of gradient underestimation. We test optimal-scale ridges for their usefulness with FTLE fields, obtaining a negative result. However, we propose an optimization of the time parameter for a given scale of observation. Finally, an incremental method for computing smoothed flow maps is presented.
Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.
We review and discuss some different techniques for describing local dispersion properties in fluids. A recent Lagrangian diagnostics based on the finite scale Lyapunov exponent (FSLE), is presented and compared to the finite time Lyapunov exponent (FTLE), to the Okubo–Weiss (OW) and Hua–Klein (HK) criteria. We show that the OW and HK are the limiting case of the FTLE, and that the FSLE is the most efficient method for detecting the presence of cross-stream barriers. We illustrate our findings by considering two examples of geophysical interest: a kinematic model of a meandering jet, and Lagrangian tracers advected by stratospheric circulation.
The Finite-time Lyapunov Exponent (FTLE) is a measure for the rate of separation of particles in time-dependent flow fields. It provides a valuable tool for the analysis of unsteady flows. Commonly it is defined based on the flow map, analyzing the separation of trajectories of nearby particles over a finite-time span. This paper proposes a localized definition of the FTLE using the Jacobian matrix along a pathline as generator of the separation. The localized FTLE (L-FTLE) definition makes only use of flow properties along the pathline. A fast computation algorithm is presented that efficiently reuses FTLE values from previous time steps, following an idea similar to FastLIC. The properties of L-FTLE are analyzed with focus on the sensitivity to the parameters of the algorithm. It is further compared to the flow map based version under consideration of robustness to noise.
The motion of fluid parcels in a two-dimensional kinematic model of a meandering jet is investigated using Melnikov's method. The study is motivated by a recent analysis of float trajectories in the Gulf Stream. The results indicate that the efficiency of cross-jet exchange induced by fluctuating meander amplitudes depends strongly on the frequency of the fluctuations. For high frequencies (= or 0.04 cpd), exchange between the core of the jet and regions of 'trapped' fluid moving with the meander is preferred, while for low frequencies (= or 0.04 cpd), exchange between the 'trapped' fluid and the slow-moving fluid surrounding the jet is preferred. Propagating waves superimposed on the meandering jet can efficiently cause exchange between regimes when their phase speeds roughly match the basic flow velocities along the regime boundaries. Numerical results suggest that exchange across the center of the jet is less efficient than exchange between adjacent regimes so that the meandering jet will tend to stir fluid along each of its sides but preserve gradients across the jet core. (A)
The first Lagrangian Analysis and Predictability of Coastal and Ocean Dynamics (LAPCOD) meeting took place in Ischia. Italy, 2-6 October 2000. The material presented at LAPCOD 2000 indicated both a maturing of Lagrangian-based observing systems and the development of new analysis and assimilation techniques for Lagrangian data. This summary presents a review of the state-of-the-art technology in Lagrangian exploration of oceanic and coastal waters that was presented at the meeting.
Much of atmospheric and oceanic transport is associated with coherent
structures. Lagrangian methods are emerging as optimal tools for their
identification and analysis. An important Lagrangian technique which is
starting to be widely used in oceanography is that of Finite-Size Lyapunov
Exponents (FSLEs). Despite this growing relevance there are still many open
questions concerning the reliability of the FSLEs in order to analyse the ocean
dynamics. In particular, it is still unclear how robust they are when
confronted with real data. In this paper we analyze the effect on this
Lagrangian technique of the two most important effects when facing real data,
namely noise and dynamics of unsolved scales. Our results, using as a
benchmarch data from a primitive numerical model of the Mediterranean Sea, show
that even when some dynamics is missed the FSLEs results still give an accurate
picture of the oceanic transport properties.
The term "Lagrangian coherent structure" (LCS) is normally used to describe numerically detected structures whose properties are similar to those of stable and unstable manifolds of hyperbolic trajectories. The latter structures are invariant curves, i.e., material curves of fluid that serve as transport barriers. In this paper we use the term LCS to describe a different type of structure whose properties are similar to those of invariant tori in certain classes of two-dimensional incompressible flows. Like stable and unstable manifolds, invariant tori are invariant curves that serve as transport barriers. There are many differences, however, between traditional LCSs and invariant-tori-like LCSs. These differences are discussed with an emphasis on numerical techniques that can be used to identify invariant-tori-like LCSs. Structures of this type are often present in geophysical flows where zonal jets are present. A prime example of an invariant-torus-like LCS is the transport barrier near the core of the polar night jet in the Earth's lower and middle stratospheres in the austral winter and early spring; this is the barrier that traps ozone-depleted air inside the ozone hole. This example is investigated using both a simple analytically prescribed flow and a velocity field produced by a general circulation model of the Earth's atmosphere.
This paper presents new efficient methods for computing finite-time Lyapunov exponent (FTLE) fields in unsteady flows. The methods approximate the particle flow map, eliminating redundant particle integrations in neighboring flow map calculations. Two classes of flow map approximations are investigated based on composition of intermediate flow maps; unidirectional approximation constructs a time-T map by composing a number of smaller time-h maps, while bidirectional approximation constructs a flow map by composing both positive- and negative-time maps. The unidirectional method is shown to be fast and accurate, although it is memory intensive. The bidirectional method is also fast and uses significantly less memory; however, it is prone to error which is large in regions where the opposite-time FTLE field is large, rendering it unusable. The algorithms are implemented and compared on three example fluid flows: a double gyre, a low Reynolds number pitching flat plate, and an unsteady ABC flow.
7 pages, 6 figures.-- AGU Index terms: 4568, 3220, 3240, 4255, 4572.-- ArXiv pre-print available: http://arxiv.org/abs/nlin/0404041 We characterize horizontal mixing and transport structures in the surface circulation of the Mediterranean Sea, as obtained from a primitive equation circulation model. We calculate the Finite Size Lyapunov Exponents (FSLEs) of the velocity data set, which gives a direct measure of the local stirring. By proper election of the FSLE parameters, we focus on the mesoscale structures, locating a number of vortices embedded in an intricate network of high-stretching lines. These lines control transport in the system. At the edge of the vortices, a dense tangle of line intersections appears, identifying strong mixing. The spatial distribution of FSLEs, averaged over one year, allows to classify areas in the Mediterranean basin according by their mixing activity. The space average of FSLEs on selected geographical regions gives a measure for quantifying and comparing the mixing seasonal variability. We acknowledge financial support from MCyT of Spain and FEDER under projects REN2001-0802-C02- 01/MAR (IMAGEN) and BFM2000-1108 (CONOCE). C.L. is a Ramón y Cajal research fellow (MCyT of Spain).
The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for Medium-Range Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another (still nonrigorous) alternative is the finite-size Lyapunov exponent (FSLE) which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures (ECS) in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECS-like barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows. (c) 2002 American Institute of Physics.
This paper presents a method for filtered ridge extraction based on adaptive mesh refinement. It is applicable in situations where the underlying scalar field can be refined during ridge extraction. This requirement is met by the concept of Lagrangian coherent structures which is based on trajectories started at arbitrary sampling grids that are independent of the underlying vector field. The Lagrangian coherent structures are extracted as ridges in finite Lyapunov exponent fields computed from these grids of trajectories. The method is applied to several variants of finite Lyapunov exponents, one of which is newly introduced. High computation time due to the high number of required trajectories is a main drawback when computing Lyapunov exponents of 3-dimensional vector fields. The presented method allows a substantial speed-up by avoiding the seeding of trajectories in regions where no ridges are present or do not satisfy the prescribed filter criteria such as a minimum finite Lyapunov exponent.
This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with
the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence.
The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.
We present a framework for different approaches to finite-time Lyapunov exponent (FTLE) computation for 2D vector fields, based on the advection of seeding circles. On the one hand it unifies the popular flow map approach with techniques based on the evaluation of distinguished trajectories, such as renormalization. On the other hand it allows for the exploration of their order of approximation (first-order approximation representing the flow map gradient). Using this framework, we derive a measure for nonlinearity of the flow map, that brings us to the definition of a new FTLE approach. We also show how the nonlinearity measure can be used as a criterion for flow map refinement for more accurate FTLE computation, and we demonstrate that ridge extraction in supersampled FTLE leads to superior ridge quality.
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.
Recent observations of fluid parcel pathways in the Gulf Stream using isopycnal RAFOS floats revealed a striking pattern of cross-stream and vertical motion associated with meanders (Bower and Rossby 1989). In an attempt to explain the observed pattern, a two-dimensional kinematic model of a meandering jet has been developed which enables examination of the relationship between streamfunction patterns and fluid parcel trajectories. The streamfunction fields are displayed in a reference frame moving with the wave pattern so motions of fluid parcels relative to the jet can be seen more easily.
The results suggest that the observed pattern of cross-stream motion results primarily from the downstream phase propagation of meanders. The model successfully reproduces several of the most distinctive features of the float observations: 1 ) entrainment of fluid into the Gulf Stream occurs at the leading edges of meander extrema while detrainment takes place at the trailing edges; 2) exchange between the Gulf Stream and its surroundings increases with a) increasing depth, b) increasing meander amplitude, and c) increasing wave phase speed.
Transport calculations from the model streamfunction fields indicate that for typical phase speeds (10 km d⁻¹) and amplitudes (50 km), roughly 90% of the fluid in the surface layers of the Gulf Stream flows downstream in the jet while 10% continuously recirculates into the surroundings. In the deep main thermocline, where downstream speeds are less, only about 40% of the fluid is retained in the jet and 60% is trapped in the recirculating cells. It is concluded that this simple kinematic mechanism could lead to cross-stream mixing of fluid parcels, especially in the deeper layers of the Gulf Stream.
A set of differential equations for the eigenvalues and eigenvectors of
the stability matrix of a dynamical system, as well as for the Lyapunov
exponents and the corresponding eigenvectors, is derived. The rate of
convergence of the Lyapunov eigenvectors is shown to be exponential. The
eigenvectors of the stability matrix can be grouped into sets, each
spanning a subspace which converges at an exponential rate. It is
demonstrated that, generically, the real parts of the eigenvalues of the
stability matrix equal the corresponding Lyapunov exponents. This
statement has been tested numerically. The values of the Lyapunov
exponents, μi, are shown to be related to the
corresponding finite time values of the Lyapunov exponents (e.g. those
deduced from a finite time numerical simulation), μi(t),
by: μi(t) = μi + (bi +
ξi(t))/t. The bi's are constants and
ξi(t) are ``noise'' terms of zero mean. This observation
leads to a method of extrapolation, which has been used to predict
Lyapunov exponents from a finite amount of data. It is shown that the
use of the standard (numerical) methods to compute Lyapunov exponents
introduces an error ai/t in the value of μi(t),
where the ai's are constants. Thus the standard method has a
rate of convergence which is the same as that of the exact
μi(t)'s. Finally, we have shown how one can compute the
eigenvectors associated with each of the eigenvalues of the stability
matrix as well as the Lyapunov eigenvectors.
Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology-based and topology-inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.
We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.
We discuss the effects of finite perturbations in fully developed turbulence by introducing a measure of the chaoticity degree associated to a given scale of the velocity field. This allows one to determine the predictability time for noninfinitesimal perturbations, generalizing the usual concept of maximum Lyapunov exponent. We also determine the scaling law for our indicator in the framework of the multifractal approach. We find that the scaling exponent is not sensitive to intermittency corrections, but is an invariant of the multifractal models. A numerical test of the results is performed in the shell model for the turbulent energy cascade.
For two-dimensional velocity fields defined on finite time intervals, we derive an analytic condition that can be used to determine numerically the location of uniformly hyperbolic trajectories. The conditions of our main theorem will be satisfied for typical velocity fields in fluid dynamics where the deformation rate of coherent structures is slower than individual particle speeds. We also propose and test a simple numerical algorithm that isolates uniformly finite-time hyperbolic sets in such velocity fields. Uniformly hyperbolic sets serve as the key building blocks of Lagrangian mixing geometry in applications. (c) 2000 American Institute of Physics.
High-frequency (HF) radar technology produces detailed velocity maps near the surface of estuaries and bays. The use of velocity data in environmental prediction, nonetheless, remains unexplored. In this paper, we uncover a striking flow structure in coastal radar observations of Monterey Bay, along the California coastline. This complex structure governs the spread of organic contaminants, such as agricultural runoff which is a typical source of pollution in the bay. We show that a HF radar-based pollution release scheme using this flow structure reduces the impact of pollution on the coastal environment in the bay. We predict the motion of the Lagrangian flow structures from finite-time Lyapunov exponents of the coastal HF velocity data. From this prediction, we obtain optimal release times, at which pollution leaves the bay most efficiently.
User guide for a modesplit σ-coordinate ocean model
- J Berntsen
Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents
- F Ovidio
- V Fernández
- E Hernández-García
- C López