Journal articles on the topic 'Finite-Scale Lyapunov Exponent'
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HAN, R., M. LEMM, and W. SCHLAG. "Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base." Ergodic Theory and Dynamical Systems 40, no. 10 (April 17, 2019): 2788–853. http://dx.doi.org/10.1017/etds.2019.19.
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We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.
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ANDRES, DANIELA SABRINA, DANIEL CERQUETTI, and MARCELO MERELLO. "FINITE DIMENSIONAL STRUCTURE OF THE GPI DISCHARGE IN PATIENTS WITH PARKINSON'S DISEASE." International Journal of Neural Systems 21, no. 03 (June 2011): 175–86. http://dx.doi.org/10.1142/s0129065711002778.
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Stochastic systems are infinitely dimensional and deterministic systems are low dimensional, while real systems lie somewhere between these two limit cases. If the calculation of a low (finite) dimension is in fact possible, one could conclude that the system under study is not purely random. In the present work we calculate the maximal Lyapunov exponent from interspike intervals time series recorded from the internal segment of the Globus Pallidusfrom patients with Parkinson's disease. We show the convergence of the maximal Lyapunov exponent at a dimension equal to 7 or 8, which is therefore our estimation of the embedding dimension for the system. For dimensions below 7 the observed behavior is what would be expected from a stochastic system or a complex system projecting onto lower dimensional spaces. The maximal Lyapunov exponent did not show any differences between tremor and akineto-rigid forms of the disease. However, it did decay with the value of motor Unified Parkinson's Disease Rating Scale -OFF scores. Patients with a more severe disease (higher UPDRS-OFF score) showed a lower value of the maximal Lyapunov exponent. Taken together, both indexes (the maximal Lyapunov exponent and the embedding dimension) remark the importance of taking into consideration the system's non-linear properties for a better understanding of the information transmission in the basal ganglia.
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Siqueira, L., and B. Kirtman. "Nonlinear dynamics approach to the predictability of the Cane–Zebiak coupled ocean–atmosphere model." Nonlinear Processes in Geophysics 21, no. 1 (January 29, 2014): 155–63. http://dx.doi.org/10.5194/npg-21-155-2014.
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Abstract. The predictability of the Cane–Zebiak coupled ocean–atmosphere model is investigated using nonlinear dynamics analysis. Newer theoretical concepts are applied to the coupled model in order to help quantify maximal prediction horizons for finite amplitude perturbations on different scales. Predictability analysis based on the maximum Lyapunov exponent considers infinitesimal perturbations, which are associated with errors in the smallest fastest-evolving scales of motion. However, these errors become irrelevant for the predictability of larger scale motions. In this study we employed finite-size Lyapunov exponent analysis to assess the predictability of the Cane–Zebiak coupled ocean–atmosphere model as a function of scale. We demonstrate the existence of fast and slow timescales, as noted in earlier studies, and the expected enhanced predictability of the anomalies on large scales. The final results and conclusions clarify the applicability of these new methods to seasonal forecasting problems.
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BozorgMagham, A. E., S. D. Ross, and D. G. Schmale. "Local finite-time Lyapunov exponent, local sampling and probabilistic source and destination regions." Nonlinear Processes in Geophysics 22, no. 6 (November 11, 2015): 663–77. http://dx.doi.org/10.5194/npg-22-663-2015.
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Abstract. The finite-time Lyapunov exponent (FTLE) is a powerful Lagrangian concept widely used for describing large-scale flow patterns and transport phenomena. However, field experiments usually have modest scales. Therefore, it is necessary to bridge the gap between the concept of FTLE and field experiments. In this paper, two independent observations are discussed: (i) approximation of the local FTLE time series at a fixed location as a function of known distances between the destination (or source) points of released (or collected) particles and local velocity, and (ii) estimation of the distances between the destination (or source) points of the released (or collected) particles when consecutive release (or sampling) events are performed at a fixed location. These two observations lay the groundwork for an ansatz methodology that can practically assist in field experiments where consecutive samples are collected at a fixed location, and it is desirable to attribute source locations to the collected particles, and also in planning of optimal local sampling of passive particles for maximal diversity monitoring of atmospheric assemblages of microorganisms. In addition to deterministic flows, the more realistic case of unresolved turbulence and low-resolution flow data that yield probabilistic source (or destination) regions are studied. It is shown that, similar to deterministic flows, Lagrangian coherent structures (LCS) and local FTLE can describe the separation of probabilistic source (or destination) regions corresponding to consecutively collected (or released) particles.
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BozorgMagham, A. E., S. D. Ross, and D. G. Schmale III. "Local finite time Lyapunov exponent, local sampling and probabilistic source and destination regions." Nonlinear Processes in Geophysics Discussions 2, no. 3 (May 28, 2015): 903–37. http://dx.doi.org/10.5194/npgd-2-903-2015.
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Abstract. The time-varying finite time Lyapunov exponent (FTLE) is a powerful Lagrangian concept widely used for describing large-scale flow patterns and transport phenomena. However, field experiments usually have modest scales. Therefore, it is necessary to bridge between the powerful concept of FTLE and (local) field experiments. In this paper a new interpretation of the local FTLE, the time series of a FTLE field at a fixed location, is proposed. This concept can practically assist in field experiments where samples are collected at a fixed location and it is necessary to attribute long distance transport phenomena and location of source points to the characteristic variation of the sampled particles. Also, results of this study have the potential to aid in planning of optimal local sampling of passive particles for maximal diversity monitoring of assemblages of microorganisms. Assuming a deterministic flow field, one can use the proposed theorem to (i) estimate the differential distances between the source (or destination) points of the collected (or released) particles when consecutive sampling (or releasing) is performed at a fixed location, (ii) estimate the local FTLE as a function of known differential distances between the source (or destination) points. In addition to the deterministic flows, the more realistic case of unresolved turbulence and low resolution flow data that yield the probabilistic source (or destination) regions are studied. It is shown that similar to deterministic flows, Lagrangian coherent structures (LCS) separate probabilistic source (or destination) regions corresponding to consecutive collected (or released) particles.
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Hidayat, A. M., U. Efendi, R. H. Virgianto, and H. A. Nugroho. "Examining optimum prediction time of rainfall dynamics based on chaotic perspective at different temporal scales: a case study in Bojonegoro, Indonesia." IOP Conference Series: Earth and Environmental Science 893, no. 1 (November 1, 2021): 012024. http://dx.doi.org/10.1088/1755-1315/893/1/012024.
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Abstract As the driving force of the hydrological system, rain has severe impact when dealing with petroleum mining activities, especially in protecting assets and safety. Rainfall has high variability, both spatial and temporal (chaotic data). Due to this reason, ones can only create long-range prediction using the stochastic method. Here we use the Lyapunov exponent to analyze the nonlinear pattern of rainfall dynamics. This method is useful for identifying chaotic deportment in rainfall data. This study uses rainfall data for six years obtained from one of the largest petroleum mining sites in Bojonegoro, Indonesia. Rainfall dynamics have been analyzed on three different time scales, namely daily data, 5-day, and 10-day. The time delay (τ) was obtained by using the Average Mutual Information (AMI) method for the three-rainfall series (3, 2, 3, respectively). The observed rainfall data in Bojonegoro show signs of chaos as the finite correlation dimensions (m) attain values about 4 for all time scales. The maximum Lyapunov exponent λmax for each of three-rainfall series in Bojonegoro is 0.111, 0.057, 0.062, respectively. These values were analyzed to find the optimum prediction time of rainfall occurrence to perform better forecasting. The result shows that the optimum range of prediction time for daily, 5-day, and 10-day have 9, 18, and 16 times longer than their temporal scale.
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Lacorata, G., E. Aurell, and A. Vulpiani. "Drifter dispersion in the Adriatic Sea: Lagrangian data and chaotic model." Annales Geophysicae 19, no. 1 (January 31, 2001): 121–29. http://dx.doi.org/10.5194/angeo-19-121-2001.
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Abstract. We analyze characteristics of drifter trajectories from the Adriatic Sea with recently introduced nonlinear dynamics techniques. We discuss how in quasi-enclosed basins, relative dispersion as a function of time, a standard analysis tool in this context, may give a distorted picture of the dynamics. We further show that useful information may be obtained by using two related non-asymptotic indicators, the Finite-Scale Lyapunov Exponent (FSLE) and the Lagrangian Structure Function (LSF), which both describe intrinsic physical properties at a given scale. We introduce a simple chaotic model for drifter motion in this system, and show by comparison with the model that Lagrangian dispersion is mainly driven by advection at sub-basin scales until saturation sets in.Key words. Oceanography: General (marginal and semi-closed seas) – Oceanography: Physical (turbulence, diffusion, and mixing processes; upper ocean processes)
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El Hassan, Mouhammad, and David S. Nobes. "Experimental Investigation of the Vortex Dynamics in Circular Jet Impinging on Rotating Disk." Fluids 7, no. 7 (July 1, 2022): 223. http://dx.doi.org/10.3390/fluids7070223.
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A circular jet impinging perpendicularly onto a rotating disk is studied in order to understand the influence of centrifugal forces on the radial wall jet. Time-resolved Particle Image Velocimetry (TR-PIV) measurements are conducted in different jet regions in order to investigate the flow physics of the large-scale vortical structures and the boundary layer development on the impinging wall for both stationary and rotating impinging disks. The Reynolds number is ReD = 2480, the orifice-to-plate distance H = 4D (D is the jet-orifice diameter) and the rotation rate is 200 RPM. It is found that the rotation of the impinging wall results in strong centrifugal effects, which affect different regions of the jet. Both radial velocity profiles and turbulence intensity distributions show different behavior when comparing the stationary and rotating cases. Finite Time Lyapunov Exponent (FTLE) analysis is implemented to describe the time-resolved behavior of the large-scale vortical structures and flow separation.
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Liu, Jian, and Xiangzhe Zhu. "Chaotic mixing analysis of a novel single-screw extruder with a perturbation baffle by the finite-time Lyapunov exponent method." Journal of Polymer Engineering 39, no. 3 (February 25, 2019): 287–99. http://dx.doi.org/10.1515/polyeng-2018-0037.
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Abstract The single-screw extruder with a perturbation baffle is a novel piece of equipment for polymer processing, in which the polymer melts undergo complex chaotic mixing. In this paper, from a new Lagrangian perspective, the fluid transporting mechanism in chaotic flow of the unwound screw channel was analyzed based on the finite element method. Firstly, two-dimensional velocity distributions in the unwound screw channel were calculated based on the mesh superposition technique. Fluid particle evolution processes in the extruder were tracked based on the fourth-order Runge-Kutta scheme. The numerical method used in this paper was validated by grid independence and experiments obtained from literature. Moreover, the finite-time Lyapunov exponent (FTLE) and Poincaré sections were adopted to discuss the chaotic mixing in the novel single-screw extruder. The effects of baffle width and height on the manifold structures in the flow dynamic system were analyzed. The results show that the homoclinic point of the manifold structure can give rise to chaotic mixing in the single-screw extruder. The height of the baffle is an important parameter to control the chaotic strength. In a way, increasing the height of the baffle can enlarge the kink scale and increase the stretching and folding actions, which results in the decrease of regular regions and an increase of the mixing efficiency in the single-screw extruder.
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Lacorata, Guglielmo, Erik Aurell, Bernard Legras, and Angelo Vulpiani. "Evidence for a k−5/3 Spectrum from the EOLE Lagrangian Balloons in the Low Stratosphere." Journal of the Atmospheric Sciences 61, no. 23 (December 1, 2004): 2936–42. http://dx.doi.org/10.1175/jas-3292.1.
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Abstract The EOLE experiment is revisited to study turbulent processes in the lower stratosphere circulation from a Lagrangian viewpoint and to resolve a discrepancy on the slope of the atmospheric energy spectrum between the work of Morel and Larchevêque and recent studies using aircraft data. Relative dispersion of balloon pairs is studied by calculating the finite-scale Lyapunov exponent, an exit-time-based technique that is particularly efficient in cases in which processes with different spatial scales are interfering. The main goal is to reconciliate the EOLE dataset with recent studies supporting a k−5/3 energy spectrum in the 100–1000-km range. The results also show exponential separation at smaller scales, with a characteristic time of order 1 day, and agree with the standard diffusion of about 107 m2 s−1 at large scales. A remaining question is the origin of a k−5/3 spectrum in the mesoscale range between 100 and 1000 km.
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Berti, Stefano, Francisco Alves Dos Santos, Guglielmo Lacorata, and Angelo Vulpiani. "Lagrangian Drifter Dispersion in the Southwestern Atlantic Ocean." Journal of Physical Oceanography 41, no. 9 (September 1, 2011): 1659–72. http://dx.doi.org/10.1175/2011jpo4541.1.
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Abstract In the framework of Monitoring by Ocean Drifters (MONDO) project, a set of Lagrangian drifters were released in proximity of the Brazil Current, the western branch of the subtropical gyre in the South Atlantic Ocean. The experimental strategy of deploying part of the buoys in clusters offers the opportunity to examine relative dispersion on a wide range of scales. Adopting a dynamical systems approach, the authors focus their attention on scale-dependent indicators, like the finite-scale Lyapunov exponent (FSLE) and the finite-scale (mean square) relative velocity (FSRV) between two drifters as a function of their separation and compare them with classic time-dependent statistical quantities like the mean-square relative displacement between two drifters and the effective diffusivity as functions of the time lag from the release. The authors find that, dependently on the given observable, the quasigeostrophic turbulence scenario is overall compatible with their data analysis, with discrepancies from the expected behavior of 2D turbulent trajectories likely to be ascribed to the nonstationary and nonhomogeneous characteristics of the flow, as well as to possible ageostrophic effects. Submesoscale features of ~O(1) km are considered to play a role, to some extent, in determining the properties of relative dispersion as well as the shape of the energy spectrum. The authors also present numerical simulations of an ocean general circulation model (OGCM) of the South Atlantic and discuss the comparison between experimental and model data about mesoscale dispersion.
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de Divitiis, Nicola. "Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review." Entropy 21, no. 5 (May 23, 2019): 520. http://dx.doi.org/10.3390/e21050520.
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This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference.
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Pérez-Muñuzuri, V., and F. Huhn. "The role of mesoscale eddies time and length scales on phytoplankton production." Nonlinear Processes in Geophysics 17, no. 2 (March 31, 2010): 177–86. http://dx.doi.org/10.5194/npg-17-177-2010.
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Abstract. Horizontal mixing has been found to play a crucial role in the development of spatial plankton structures in the ocean. We study the influence of time and length scales of two different horizontal two-dimensional (2-D) flows on the growth of a single phytoplankton patch. To that end, we use a coupled model consisting of a standard three component ecological NPZ model and a flow model able to mimic the mesoscale structures observed in the ocean. Two hydrodynamic flow models are used: a flow based on Gaussian correlated noise, for which the Eulerian length and time scales can be easily controlled, and a multiscale velocity field derived from altimetry data in the North Atlantic ocean. We find the optimal time and length scales for the Gaussian flow model favouring the plankton spread. These results are used for an analysis of a more realistic altimetry flow. We discuss the findings in terms of the time scale of the NPZ model, the qualitative interaction of the flow with the reaction front and a Finite-Time Lyapunov Exponent analysis.
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Yang, Huijun. "The Central Barrier, Asymmetry and Random Phase in Chaotic Transport and Mixing by Rossby Waves in a Jet." International Journal of Bifurcation and Chaos 08, no. 06 (June 1998): 1131–52. http://dx.doi.org/10.1142/s0218127498000905.
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The central barrier, asymmetry and random perturbation in transport and mixing by Rossby waves in a jet were investigated by simple kinematic model. Two complementary methods were used: A high-resolution Lagrangian Field Advection Model (FAM) and a finite-time Lyapunov exponent analysis. The present study revealed the following: (1) A central barrier can be formed in two Rossby waves without shear flow as well as in a jet, (2) the central barrier may occur in the region with maximum jet speed relative to the phase speed of the traveling wave, whereas the chaotic mixing most likely occurs near the critical lines; the central barrier widens as the phase speed of traveling waves relative to the jet speed increases, (3) asymmetry of wave-breaking is directly related to asymmetry of the critical line location in a jet, (4) the central barrier survives small random perturbations, (5) global bifurcation from a homoclinic orbit to a heteroclinic orbit and global chaos are two main mechanisms for the central barrier destruction. The results suggest that the small scale motions and random processes may not significantly affect the major character of Lagrangian transport and mixing by large-scale geophysical flow. Also potential vorticity mixing provides a unique kinematic and dynamic view of many features of the geophysical flow.
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Zhu, Xiang, Ying Tong, and Yue Hu. "Chaotic Manifold Analysis of Four-Screw Extruders Based on Lagrangian Coherent Structures." Materials 11, no. 11 (November 14, 2018): 2272. http://dx.doi.org/10.3390/ma11112272.
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The four-screw extruder (FSE) is a novel equipment for polymer processing. In this paper, from a new viewpoint of Lagrangian coherent structures (LCS), two-dimensional fluid transport and chaotic mixing characteristics within three kinds of novel industrial FSEs are explored based on LCS to better understand the flow and mixing natures in the FSEs. Firstly, based on the finite-time invariant manifold theory, the finite-time Lyapunov exponent (FTLE) and LCS of FSEs are calculated by considering the different initial time. Hyperbolic LCSs from the FTLE maps are adopted to identify chaotic mixing manifolds in FSEs. Moreover, particle tracking and Poincaré sections are used to illustrate the different fluid motions in the above three isolated regions. Finally, the effects of relative rotating directions and layout of four screws on the chaotic manifolds in FESs are discussed in order to enhance local mixing performance. Furthermore, quantitative mixing measures, such as the segregation scale, logarithmic of stretching, and mean-time mixing efficiency are employed to compare the mixing efficiencies in three kinds of FSEs. The results show that the relative rotating directions and positions of four screws can change the chaotic manifolds and increase mixing performance in local poor mixing regions. FTLE and LCS analysis are helpful to better understand the chaotic mixing nature in the novel screw extruders.
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Perez, Gabriel M. P., Pier Luigi Vidale, Nicholas P. Klingaman, and Thomas C. M. Martin. "Atmospheric convergence zones stemming from large-scale mixing." Weather and Climate Dynamics 2, no. 2 (June 9, 2021): 475–88. http://dx.doi.org/10.5194/wcd-2-475-2021.
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Abstract. Organised cloud bands are important features of tropical and subtropical rainfall. These structures are often regarded as convergence zones, alluding to an association with coherent atmospheric flow. However, the flow kinematics is not usually taken into account in classification methods for this type of event, as large-scale lines are rarely evident in instantaneous diagnostics such as Eulerian convergence. Instead, existing convergence zone definitions rely on heuristic rules of shape, duration and size of cloudiness fields. Here we investigate the role of large-scale turbulence in shaping atmospheric moisture in South America. We employ the finite-time Lyapunov exponent (FTLE), a metric of deformation among neighbouring trajectories, to define convergence zones as attracting Lagrangian coherent structures (LCSs). Attracting LCSs frequent tropical and subtropical South America, with climatologies consistent with the South Atlantic Convergence Zone (SACZ), the South American Low-Level Jet (SALLJ) and the Intertropical Convergence Zone (ITCZ). In regions under the direct influence of the ITCZ and the SACZ, rainfall is significantly positively correlated with large-scale mixing measured by the FTLE. Attracting LCSs in south and southeast Brazil are associated with significant positive rainfall and moisture flux anomalies. Geopotential height composites suggest that the occurrence of attracting LCSs in these regions is related with teleconnection mechanisms such as the Pacific–South Atlantic. We believe that this kinematical approach can be used as an alternative to region-specific convergence zone classification algorithms; it may help advance the understanding of underlying mechanisms of tropical and subtropical rain bands and their role in the hydrological cycle.
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Linden, Nathaniel J., Dennis R. Tabuena, Nicholas A. Steinmetz, William J. Moody, Steven L. Brunton, and Bingni W. Brunton. "Go with the FLOW: visualizing spatiotemporal dynamics in optical widefield calcium imaging." Journal of The Royal Society Interface 18, no. 181 (August 2021): 20210523. http://dx.doi.org/10.1098/rsif.2021.0523.
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Widefield calcium imaging has recently emerged as a powerful experimental technique to record coordinated large-scale brain activity. These measurements present a unique opportunity to characterize spatiotemporally coherent structures that underlie neural activity across many regions of the brain. In this work, we leverage analytic techniques from fluid dynamics to develop a visualization framework that highlights features of flow across the cortex, mapping wavefronts that may be correlated with behavioural events. First, we transform the time series of widefield calcium images into time-varying vector fields using optic flow. Next, we extract concise diagrams summarizing the dynamics, which we refer to as FLOW (flow lines in optical widefield imaging) portraits . These FLOW portraits provide an intuitive map of dynamic calcium activity, including regions of initiation and termination, as well as the direction and extent of activity spread. To extract these structures, we use the finite-time Lyapunov exponent technique developed to analyse time-varying manifolds in unsteady fluids. Importantly, our approach captures coherent structures that are poorly represented by traditional modal decomposition techniques. We demonstrate the application of FLOW portraits on three simple synthetic datasets and two widefield calcium imaging datasets, including cortical waves in the developing mouse and spontaneous cortical activity in an adult mouse.
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Haza, Angelique C., Tamay M. Özgökmen, Annalisa Griffa, Andrew C. Poje, and M. Pascale Lelong. "How Does Drifter Position Uncertainty Affect Ocean Dispersion Estimates?" Journal of Atmospheric and Oceanic Technology 31, no. 12 (December 1, 2014): 2809–28. http://dx.doi.org/10.1175/jtech-d-14-00107.1.
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Abstract To develop methodologies to maximize the information content of Lagrangian data subject to position errors, synthetic trajectories produced by both a large-eddy simulation (LES) of an idealized submesoscale flow field and a high-resolution Hybrid Coordinate Ocean Model simulation of the North Atlantic circulation are analyzed. Scale-dependent Lagrangian measures of two-particle dispersion, mainly the finite-scale Lyapunov exponent [FSLE; λ(δ)], are used as metrics to determine the effects of position uncertainty on the observed dispersion regimes. It is found that the cumulative effect of position uncertainty on λ(δ) may extend to scales 20–60 times larger than the position uncertainty. The range of separation scales affected by a given level of position uncertainty depends upon the slope of the true FSLE distribution at the scale of the uncertainty. Low-pass filtering or temporal subsampling of the trajectories reduces the effective noise amplitudes at the smallest spatial scales at the expense of limiting the maximum computable value of λ. An adaptive time-filtering approach is proposed as a means of extracting the true FSLE signal from data with uncertain position measurements. Application of this filtering process to the drifters with the Argos positioning system released during the LatMix: Studies of Submesoscale Stirring and Mixing (2011) indicates that the measurement noise dominates the dispersion regime in λ for separation scales δ < 3 km. An expression is provided to estimate position errors that can be afforded depending on the expected maximum λ in the submesoscale regime.
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Rudko, M. V., I. V. Kamenkovich, and D. S. Nolan. "Stability of Baroclinic Vortices on the β Plane and Implications for Transport." Journal of Physical Oceanography 46, no. 11 (November 2016): 3245–62. http://dx.doi.org/10.1175/jpo-d-16-0067.1.
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AbstractThis paper explores stability of and transport by baroclinic vortices on the β plane using a two-layer, quasigeostrophic model. The study adapts a wave–mean flow formalism and examines interactions between the axisymmetric flow (“the vortex”) and residuals (“the waves”). Unlike baroclinically unstable vortices on the f plane, such vortices on the β plane can be also vulnerable to barotropic instability as revealed by the globally integrated energy balance analysis. The spatial structure of energy fluxes shows the energy leakage inside the vortex core when its breakdown occurs. Mixing by stable and unstable vortical flows is quantified through the computation of finite-time Lyapunov exponent (FTLE) maps. Depending on the strength of wave radiation, the upper-layer FTLE maps of stable vortices show either an annulus or volute ring of vigorous mixing inside the vortex interior. This ring region is disrupted when the vortex becomes unstable. Both stable and unstable vortices show the wavy patterns of FTLE in the near and far fields. Despite the fact that the initial vortex resides in the top layer only, significant FTLE patterns are observed in the deep layer at later times. Lagrangian analysis of the vortex-induced change of large-scale tracer gradient demonstrates significant effects of vortex instability in the top layer and the importance of the wavelike anomalies in the bottom layer.
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Kim, Youngwoo, Chanhee Moon, Omid Nematollahi, Hyun Dong Kim, and Kyung Chun Kim. "Time-Resolved PIV Measurements and Turbulence Characteristics of Flow Inside an Open-Cell Metal Foam." Materials 14, no. 13 (June 25, 2021): 3566. http://dx.doi.org/10.3390/ma14133566.
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Open-cell metal foams are porous medium for thermo-fluidic systems. However, their complex geometry makes it difficult to perform time-resolved (TR) measurements inside them. In this study, a TR particle image velocimetry (PIV) method is introduced for use inside open-cell metal foam structures. Stereolithography 3D printing methods and conventional post-processing methods cannot be applied to metal foam structures; therefore, PolyJet 3D printing and post-processing methods were employed to fabricate a transparent metal foam replica. The key to obtaining acceptable transparency in this method is the complete removal of the support material from the printing surfaces. The flow characteristics inside a 10-pore-per-inch (PPI) metal foam were analyzed in which porosity is 0.92 while laminar flow condition is applied to inlet. The flow inside the foam replica is randomly divided and combined by the interconnected pore network. Robust crosswise motion occurs inside foam with approximately 23% bulk speed. Strong influence on transverse motion by metal foam is evident. In addition, span-wise vorticity evolution is similar to the integral time length scale of the stream-wise center plane. The span-wise vorticity fluctuation through the foam arrangement is presented. It is believed that this turbulent characteristic is caused by the interaction of jets that have different flow directions inside the metal foam structure. The finite-time Lyapunov exponent method is employed to visualize the vortex ridges. Fluctuating attracting and repelling material lines are expected to enhance the heat and mass transfer. The results presented in this study could be useful for understanding the flow characteristics inside metal foams.
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Biferale, L., C. Meneveau, and R. Verzicco. "Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence." Journal of Fluid Mechanics 754 (July 30, 2014): 184–207. http://dx.doi.org/10.1017/jfm.2014.366.
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AbstractSmall droplets in turbulent flows can undergo highly variable deformations and orientational dynamics. For neutrally buoyant droplets smaller than the Kolmogorov scale, the dominant effects from the surrounding turbulent flow arise through Lagrangian time histories of the velocity gradient tensor. Here we study the evolution of representative droplets using a model that includes rotation and stretching effects from the surrounding fluid, and restoration effects from surface tension including a constant droplet volume constraint, while assuming that the droplets maintain an ellipsoidal shape. The model is combined with Lagrangian time histories of the velocity gradient tensor extracted from direct numerical simulations (DNS) of turbulence to obtain simulated droplet evolutions. These are used to characterize the size, shape and orientation statistics of small droplets in turbulence. A critical capillary number is identified associated with unbounded growth of one or two of the droplet’s semi-axes. Exploiting analogies with dynamics of polymers in turbulence, the critical capillary number can be predicted based on the large deviation theory for the largest finite-time Lyapunov exponent quantifying the chaotic separation of particle trajectories. Also, for subcritical capillary numbers near the critical value, the theory enables predictions of the slope of the power-law tails of droplet size distributions in turbulence. For cases when the viscosities of droplet and outer fluid differ in a way that enables vorticity to decorrelate the shape from the straining directions, the large deviation formalism based on the stretching properties of the velocity gradient tensor loses validity and its predictions fail. Even considering the limitations of the assumed ellipsoidal droplet shape, the results highlight the complex coupling between droplet deformation, orientation and the local fluid velocity gradient tensor to be expected when small viscous drops interact with turbulent flows. The results also underscore the usefulness of large deviation theory to model these highly complex couplings and fluctuations in turbulence that result from time integrated effects of fluid deformations.
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de Divitiis, Nicola. "Statistics of Finite Scale Local Lyapunov Exponents in Fully Developed Homogeneous Isotropic Turbulence." Advances in Mathematical Physics 2018 (June 7, 2018): 1–12. http://dx.doi.org/10.1155/2018/2365602.
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The present work analyzes the statistics of finite scale local Lyapunov exponents of pairs of fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this statistics is here analyzed assuming that the entropy associated with the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results in an unsymmetrical uniform function in a proper interval of variation. From this PDF, we determine the relationship between average and maximum Lyapunov exponents and the longitudinal velocity correlation function. This link, which in turn leads to the closure of von Kármán–Howarth and Corrsin equations, agrees with results of previous works, supporting the proposed PDF calculation, at least for the purposes of the energy cascade main effect estimation. Furthermore, through the property that the Lyapunov vectors tend to align the direction of the maximum growth rate of trajectories distance, we obtain the link between maximum and average Lyapunov exponents in line with the previous results. To validate the proposed theoretical results, we present different numerical simulations whose results justify the hypotheses of the present analysis.
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García-Olivares, A., J. Isern-Fontanet, and E. García-Ladona. "Dispersion of passive tracers and finite-scale Lyapunov exponents in the Western Mediterranean Sea." Deep Sea Research Part I: Oceanographic Research Papers 54, no. 2 (February 2007): 253–68. http://dx.doi.org/10.1016/j.dsr.2006.10.009.
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Garaboa-Paz, Daniel, Jorge Eiras-Barca, and Vicente Pérez-Muñuzuri. "Climatology of Lyapunov exponents: the link between atmospheric rivers and large-scale mixing variability." Earth System Dynamics 8, no. 3 (September 26, 2017): 865–73. http://dx.doi.org/10.5194/esd-8-865-2017.
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Abstract. Large-scale tropospheric mixing and Lagrangian transport properties have been analyzed for the long-term period 1979–2014 in terms of the finite-time Lyapunov exponents (FTLEs). Wind field reanalyses from the European Centre for Medium-Range Weather Forecasts were used to calculate the Lagrangian trajectories of large ensembles of particles. Larger values of the interannual and intra-annual mixing variabilities highlight the El Niño Southern Oscillation, the storm track, or the Intertropical Convergence Zone among other large-scale structures. The mean baroclinic instability growth rate and the mean atmospheric river occurrence show large correlation values with the FTLE climatology as an indication of their influence on tropospheric mixing in the midlatitudes. As a case study, the role that land-falling atmospheric rivers have on large-scale tropospheric mixing and the precipitation rates observed in Saharan Morocco and the British Isles has been analyzed. The atmospheric river contribution to tropospheric mixing is found to decrease from 15 % in Saharan Morocco to less than 5 % for the UK and Ireland regions, in agreement with their contribution to precipitation that is 40 % larger in the former than in the latter region.
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Garaboa-Paz, Daniel, Nieves Lorenzo, and Vicente Pérez-Muñuzuri. "Influence of finite-time Lyapunov exponents on winter precipitation over the Iberian Peninsula." Nonlinear Processes in Geophysics 24, no. 2 (May 24, 2017): 227–35. http://dx.doi.org/10.5194/npg-24-227-2017.
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Abstract. Seasonal forecasts have improved during the last decades, mostly due to an increase in understanding of the coupled ocean–atmosphere dynamics, and the development of models able to predict the atmosphere variability. Correlations between different teleconnection patterns and severe weather in different parts of the world are constantly evolving and changing. This paper evaluates the connection between winter precipitation over the Iberian Peninsula and the large-scale tropospheric mixing over the eastern Atlantic Ocean. Finite-time Lyapunov exponents (FTLEs) have been calculated from 1979 to 2008 to evaluate this mixing. Our study suggests that significant negative correlations exist between summer FTLE anomalies and winter precipitation over Portugal and Spain. To understand the mechanisms behind this correlation, summer anomalies of the FTLE have also been correlated with other climatic variables such as the sea surface temperature (SST), the sea level pressure (SLP) or the geopotential. The East Atlantic (EA) teleconnection index correlates with the summer FTLE anomalies, confirming their role as a seasonal predictor for winter precipitation over the Iberian Peninsula.
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Waugh, Darryn W., Shane R. Keating, and Mei-Lin Chen. "Diagnosing Ocean Stirring: Comparison of Relative Dispersion and Finite-Time Lyapunov Exponents." Journal of Physical Oceanography 42, no. 7 (July 1, 2012): 1173–85. http://dx.doi.org/10.1175/jpo-d-11-0215.1.
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Abstract The relationship between two commonly used diagnostics of stirring in ocean and atmospheric flows, the finite-time Lyapunov exponents λ and relative dispersion R2, is examined for a simple uniform strain flow and ocean flow inferred from altimetry. Although both diagnostics are based on the separation of initially close particles, the two diagnostics measure different aspects of the flow and, in general, there is not a one-to-one relationship between the diagnostics. For a two-dimensional flow with time-independent uniform strain, there is a single time-independent λ, but there is a wide range of values of R2 for individual particle pairs. However, it is shown that the upper and lower limits of R2 for individual pairs, the mean value over a large ensemble of pairs, and the probability distribution function (PDF) of R2 have simple relationships with λ. Furthermore, these analytical expressions provide a reasonable approximation for the R2–λ relationship in the surface ocean flow based on geostrophic velocities derived from satellite altimeter measurements. In particular, the bimodal distribution, upper and lower bounds, and mean values from the ocean flow are similar to the analytical expressions for a uniform strain flow. How well, as well as over what integration time scale, this holds depends on the spatial and temporal variations within the ocean region being considered.
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STOCCHINO, ALESSANDRO, GIOVANNI BESIO, SONIA ANGIOLANI, and MAURIZIO BROCCHINI. "Lagrangian mixing in straight compound channels." Journal of Fluid Mechanics 675 (March 29, 2011): 168–98. http://dx.doi.org/10.1017/s0022112011000127.
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Recently Stocchino & Brocchini (J. Fluid Mech., vol. 643, 2010, p. 425 have studied the dynamics of two-dimensional (2D) large-scale vortices with vertical axis evolving in a straight compound channel under quasi-uniform flow conditions. The mixing processes associated with such vortical structures are here analysed through the results of a dedicated experimental campaign. Time-resolved Eulerian surface velocity fields, measured using a 2D particle-image velocimetry system, form the basis for a Lagrangian analysis of the dispersive processes that occur in compound channels when the controlling physical parameters, i.e. the flow depth ratio (rh) and the Froude number (Fr) are changed. Lagrangian mixing is studied by means of various approaches based either on single-particle or multiple-particle statistics (relative and absolute statistics, probability density functions (p.d.f.s) of relative displacements and finite-scale Lyapunov exponents). Absolute statistics reveal that transitional macrovortices, typical of shallow flow conditions, strongly influence the growth in time of the total absolute dispersion, after the initial ballistic regime, leading to a non-monotonic behaviour. In deep flow conditions, on the contrary, the absolute dispersion displays a monotonic growth because the generation of transitional macrovortices does not take place. In all cases an asymptotic diffusive regime is reached.Multiple-particle dynamics is controlled by rh and Fr. Different growth regimes of the relative diffusivity have been found depending on the flow conditions. This behaviour can be associated with different energy transfer processes and it is further confirmed by the p.d.f.s of relative displacements, which show a different asymptotical shape depending on the separation scales and the Froude number. Finally, an equilibrium regime is observed for all the experiments by analysing the decay of the finite-scale Lyapunov exponents with the particle separations.
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Carlu, Mallory, Francesco Ginelli, Valerio Lucarini, and Antonio Politi. "Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model." Nonlinear Processes in Geophysics 26, no. 2 (May 7, 2019): 73–89. http://dx.doi.org/10.5194/npg-26-73-2019.
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Abstract. We investigate the geometrical structure of instabilities in the two-scale Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection onto the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer timescales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a nontrivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, “mixing” their evolution into a set of vectors which simultaneously carry information on both scales. We suggest that these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.
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Resseguier, Valentin, Bertrand Chapron, and Etienne Mémin. "Effects of Smooth Divergence-Free Flows on Tracer Gradients and Spectra: Eulerian Prognosis Description." Journal of Physical Oceanography 52, no. 1 (January 2022): 53–74. http://dx.doi.org/10.1175/jpo-d-21-0014.1.
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Abstract Ocean eddies play an important role in the transport of heat, salt, nutrients, or pollutants. During a finite-time advection, the gradients of these tracers can increase or decrease, depending on a growth rate and the angle between flow gradients and initial tracer gradients. The growth rate is directly related to finite-time Lyapunov exponents. Numerous studies on mixing and/or tracer downscaling methods rely on satellite altimeter-derived ocean velocities. Filtering most oceanic small-scale eddies, the resulting smooth Eulerian velocities are often stationary during the characteristic time of tracer gradient growth. While smooth, these velocity fields are still locally misaligned, and thus uncorrelated, to many coarse-scale tracer observations amendable to downscaling [e.g., sea surface temperature (SST), sea surface salinity (SSS)]. Using finite-time advections, the averaged squared norm of tracer gradients can then only increase, with local growth rate independent of the initial coarse-scale tracer distribution. The key mixing processes are then only governed by locally uniform shears and foldings around stationary convective cells. To predict the tracer deformations and the evolution of their second-order statistics, an efficient proxy is proposed. Applied to a single velocity snapshot, this proxy extends the Okubo–Weiss criterion. For the Lagrangian-advection-based downscaling methods, it further successfully predicts the evolution of tracer spectral energy density after a finite time, and the optimal time to stop the downscaling operation. A practical estimation can then be proposed to define an effective parameterization of the horizontal eddy diffusivity. Significance Statement An analytical formalism is adopted to derive new exact and approximate relations that express the clustering of tracers transported by upper-ocean flows. This formalism bridges previous Eulerian and Lagrangian approaches. Accordingly, for slow and smooth upper-ocean flows, a rapid prognosis estimate can solely be performed using single-time velocity field observations. Well suited to satellite-altimeter measurements, it will help rapidly identify and monitor mixing regions occurring in the vicinity of ocean eddy boundaries.
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Espa, Stefania, Guglielmo Lacorata, and Gabriella Di Nitto. "Anisotropic Lagrangian Dispersion in Rotating Flows with a β Effect." Journal of Physical Oceanography 44, no. 2 (February 1, 2014): 632–43. http://dx.doi.org/10.1175/jpo-d-13-045.1.
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Abstract A detailed analysis of Lagrangian tracer dispersion is performed on datasets obtained from laboratory experiments that simulate rotating turbulence in the presence of a β effect. Compatible with the limitations of the experimental apparatus, a relatively wide range of the zonostrophy index Rβ, a parameter used to characterize flow regimes in β-plane turbulence, is explored. The considered range spans from values ~O(10−1), for which the flow is nearly isotropic, to values ~O(1), corresponding to the so-called transitional range in which the flow gradually leaves the friction-dominated regime to enter the full zonostrophic regime. The degree of anistropy and the characteristic scales of the flow have been estimated by means of a Lagrangian approach based on the reconstruction of tracer trajectories and on the measure of the finite-scale Lyapunov exponents (FSLE). The FSLE analysis allows one to identify the regimes of two-particle dispersion and to relate them to the physical parameters of the system. Moreover, a Lagrangian anisotropy index (LAI) is introduced and defined in terms of the FSLE zonal and radial components, in order to describe the onset of anisotropy and to check if it is consistent with the theoretical predictions. It is remarkable that the finite-scale dispersion rates are very sensitive to the degree of anistropy of turbulence, more so than other indicators defined in terms of Eulerian quantities. Furthermore, they offer an effective diagnostic tool of the degree of anisotropy that can be used even prior to attaining a fully developed regime of zonostrophic turbulence.
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He, Guo-Sheng, Chong Pan, Li-Hao Feng, Qi Gao, and Jin-Jun Wang. "Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat plate boundary layer." Journal of Fluid Mechanics 792 (March 3, 2016): 274–306. http://dx.doi.org/10.1017/jfm.2016.81.
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Evolution of Lagrangian coherent structures (LCS) in a flat plate boundary layer transition induced by the wake of a circular cylinder is investigated. Both hydrogen bubble visualization and particle image velocimetry (PIV) techniques are used. It is found that downstream of the cylinder, the disturbance in the boundary layer experiences a fast growth followed by a slow decay in the transition. Lagrangian coherent structures are revealed by qualitative hydrogen bubble visualizations and quantitative finite-time Lyapunov exponents (FTLE) fields derived from the PIV data. The evolution of the LCS is considered from the very beginning of the transition up to when the boundary layer becomes fully developed turbulent flow. The mean convection velocity and average inclination angle of the LCS are first extracted from the FTLE fields. The streamwise length of the low-speed streaks seems to increase, while their spanwise distance decreases in the boundary layer transition. Proper orthogonal decomposition (POD) of the PIV data shows that low-speed streaks associated with the hairpin vortices and hairpin packets are the dominant coherent structures close to the wall in the transitional and turbulent boundary layer. The POD modes also reveal a variety of scales in the turbulent boundary layer. Moreover, it is found that large-scale coherent structures can modulate the amplitude of the small-scale ones.
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Durán Colmenares, A. H., and L. Zavala Sansón. "Anisotropic Lagrangian dispersion in zonostrophic turbulence in a closed basin." Physics of Fluids 34, no. 10 (October 2022): 106605. http://dx.doi.org/10.1063/5.0105846.
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This article studies the anisotropic particle dispersion in a continuously forced, two-dimensional turbulent flow on a β-plane. The flow is immersed in a large-scale closed basin with free-slip walls. The anisotropy is analyzed in two sets of numerical experiments characterized by the magnitude of the imposed, time-dependent forcing (weak and strong). Both experiments exhibit typical features of zonostrophic turbulence: eddy motions that, on average, form alternating east–west circulation bands due to the β-effect. The dispersion anisotropy is investigated through three Lagrangian statistics calculated by zonal and meridional components: (i) relative dispersion between pairs of particles; (ii) dispersion ellipses; and (iii) finite-scale Lyapunov exponents (FSLE), also measured with particle pairs. In the experiment with weak forcing, the relative dispersion and dispersion ellipses show anisotropy with a zonal preference toward the west; however, the FSLE did not reveal significant anisotropy. In the experiment with strong forcing, the relative dispersion and dispersion ellipses show zonal anisotropy toward the west when the particles are far from the boundaries. As the particles reach the western wall and are redistributed to fill the domain, the anisotropy ceases. The FSLE show zonal anisotropy for a wide range of particle separations. The results are examined further by using no-slip boundary conditions and a rectangular domain geometry.
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Keating, Shane R., K. Shafer Smith, and Peter R. Kramer. "Diagnosing Lateral Mixing in the Upper Ocean with Virtual Tracers: Spatial and Temporal Resolution Dependence." Journal of Physical Oceanography 41, no. 8 (August 1, 2011): 1512–34. http://dx.doi.org/10.1175/2011jpo4580.1.
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Abstract Several recent studies diagnose lateral stirring and mixing in the upper ocean using altimetry-derived velocity fields to advect “virtual” particles and fields offline. However, the limited spatiotemporal resolution of altimetric maps leads to errors in the inferred diagnostics, because unresolved scales are necessarily imperfectly modeled. The authors examine a range of tracer diagnostics in two models of baroclinic turbulence: the standard Phillips model, in which dispersion is controlled by large-scale eddies, and the Eady model, where dispersion is determined by local scales of motion. These models serve as a useful best- and worst-case comparison and a valuable test of the resolution sensitivity of tracer diagnostics. The effect of unresolved scales is studied by advecting tracers using model velocity fields subsampled in space and time and comparing the derived tracer diagnostics with their “true” value obtained from the fully resolved flow. The authors find that eddy diffusivity and absolute dispersion, which are governed by large-scale dynamics, are insensitive to spatial sampling error in either flow. Measures that depend strongly on small scales, such as relative dispersion and finite-time Lyapunov exponents, are highly sensitive to spatial sampling in the Eady model. Temporal sampling error is found to have a more complicated behavior because of the onset of particle overshoot leading to scrambling of Lagrangian diagnostics. This leads to a potential restriction on the utility of raw altimetry maps for studying mixing in the upper ocean. The authors conclude that offline diagnostics of mixing in ocean flows with an energized submesoscale should be viewed with some caution.
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Li, Jiahua, Xiang Qiu, Yu Shao, Haoxuan Liu, Yuan Fu, Yizhou Tao, and Yulu Liu. "Turbulent coherent structures in channel flow with a wall-mounted hemisphere." AIP Advances 12, no. 3 (March 1, 2022): 035006. http://dx.doi.org/10.1063/5.0079605.
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Turbulent channel flows around a wall-mounted hemisphere numerically are investigated by large eddy simulation, and the Reynolds number based on the hemisphere’s diameter is 3 × 104. The statistical characteristics and turbulent structure evolution are revealed in the Eulerian frameworks and Lagrangian frameworks. The vortex identification and Dynamic Mode Decomposition (DMD) are used to study the evolution of turbulent structure in the Eulerian frameworks, and the finite-time Lyapunov exponents are applied to identify Lagrangian coherent structures (LCS) in the Lagrangian framework. It is found that the developing angle of the hairpin vortex is ∼7° at two frameworks. What is more, there are some hairpin vortices formed behind the hemisphere and some turbulent structures formed near the wall by DMD method. The correlation analysis is applied to investigate the angle variation and scale variation of turbulent structures, and it is observed that the angle of turbulent structures is negative at Y/ d ≥ 1.2 and the spanwise length scales of turbulent structures increase as it moves downstream. By studying the LCS behind a wall-mounted hemisphere, there is formation of “kink” caused by viscous interaction between some hairpin vortex legs, which is the characteristic of hairpin vortex deformation. The comparisons of statistical characteristics between Eulerian frameworks and Lagrangian frameworks are conducted by the correlation analysis, the spectrum analysis, and the structure functions.
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Rousselet, Louise, Alain de Verneil, Andrea M. Doglioli, Anne A. Petrenko, Solange Duhamel, Christophe Maes, and Bruno Blanke. "Large- to submesoscale surface circulation and its implications on biogeochemical/biological horizontal distributions during the OUTPACE cruise (southwest Pacific)." Biogeosciences 15, no. 8 (April 20, 2018): 2411–31. http://dx.doi.org/10.5194/bg-15-2411-2018.
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Abstract. The patterns of the large-scale, meso- and submesoscale surface circulation on biogeochemical and biological distributions are examined in the western tropical South Pacific (WTSP) in the context of the OUTPACE cruise (February–April 2015). Multi-disciplinary original in situ observations were achieved along a zonal transect through the WTSP and their analysis was coupled with satellite data. The use of Lagrangian diagnostics allows for the identification of water mass pathways, mesoscale structures, and submesoscale features such as fronts. In particular, we confirmed the existence of a global wind-driven southward circulation of surface waters in the entire WTSP, using a new high-resolution altimetry-derived product, validated by in situ drifters, that includes cyclogeostrophy and Ekman components with geostrophy. The mesoscale activity is shown to be responsible for counter-intuitive water mass trajectories in two subregions: (i) the Coral Sea, with surface exchanges between the North Vanuatu Jet and the North Caledonian Jet, and (ii) around 170∘ W, with an eastward pathway, whereas a westward general direction dominates. Fronts and small-scale features, detected with finite-size Lyapunov exponents (FSLEs), are correlated with 25 % of surface tracer gradients, which reveals the significance of such structures in the generation of submesoscale surface gradients. Additionally, two high-frequency sampling transects of biogeochemical parameters and microorganism abundances demonstrate the influence of fronts in controlling the spatial distribution of bacteria and phytoplankton, and as a consequence the microbial community structure. All circulation scales play an important role that has to be taken into account not only when analysing the data from OUTPACE but also, more generally, for understanding the global distribution of biogeochemical components.
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Waugh, Darryn W., Edward R. Abraham, and Melissa M. Bowen. "Spatial Variations of Stirring in the Surface Ocean: A Case Study of the Tasman Sea." Journal of Physical Oceanography 36, no. 3 (March 1, 2006): 526–42. http://dx.doi.org/10.1175/jpo2865.1.
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Abstract Stirring in the Tasman Sea is examined using surface geostrophic currents derived from satellite altimeter measurements. Calculations of the distribution of finite-time Lyapunov exponents (FTLEs) indicate that the stirring in this region is not uniform and stretching rates over 15 days vary from less than 0.02 day−1 to over 0.3 day−1. These variations occur at both small (∼10 km) and large (∼1000 km) scales and in both cases are linked to dynamical features of the flow. The small-scale variations are related to the characteristics of coherent vortex structures, and there are low FTLEs inside vortices and filaments of high FTLEs in strain-dominated regions surrounding these vortices. Regional variations in the stirring are closely related to variations in mesoscale activity and eddy kinetic energy (EKE). High values of mean FTLE occur in regions of high EKE (highest mean values of around 0.2 day−1 occur in the East Australia Current separation region) whereas small values occur in regions with low EKE (mean values around 0.03 day−1 in the east Tasman Sea). There is a compact relationship between the mean FTLEs and EKE, raising the possibility of using the easily calculated EKE to estimate the stirring. This possibility is even more intriguing because the FTLE distributions can be approximated, for the time scales considered here, by Weibull distributions with shape parameter equal to 1.6, which can be defined from the mean value alone.
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Sansón, Luis Zavala, Paula Pérez-Brunius, and Julio Sheinbaum. "Surface Relative Dispersion in the Southwestern Gulf of Mexico." Journal of Physical Oceanography 47, no. 2 (February 2017): 387–403. http://dx.doi.org/10.1175/jpo-d-16-0105.1.
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AbstractSurface dispersion properties in the southwestern Gulf of Mexico are studied by using a set of 441 drifters released during a 7-yr period and tracked for 2 months on average. The drifters have a drogue below the surface Ekman layer, so they approximately follow oceanic currents. This study follows two different approaches: First, two-particle (or pair) statistics are calculated [relative dispersion and finite-scale Lyapunov exponents (FSLEs)]. Relative dispersion estimates are consistent with theoretical dispersion regimes of two-dimensional turbulence: an exponential growth during the first 3 days, a Richardson-like regime between 3 and 20 days (in which relative dispersion grows as a power law in time), and standard dispersion (linear growth) for longer times. The FSLEs yield a power-law regime for scales between 10 and 150 km but do not detect an exponential regime for short separations (less than 10 km). Robust estimates of diffusivities based on both relative dispersion and FSLEs are provided. Second, two different dispersion scenarios are revealed by drifter trajectories and altimetric data and supported by two-particle statistics: (i) a south-to-north advection of drifters, predominantly along the western shelf of the region, and (ii) a retention of drifters during several weeks at the Bay of Campeche, the southernmost part of the Gulf of Mexico. Dominant processes that control the dispersion are the arrival of anticyclonic Loop Current eddies to the western shelf and their interaction with the semipermanent cyclonic structure in the Bay of Campeche.
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Dräger-Dietel, J., K. Jochumsen, A. Griesel, and G. Badin. "Relative Dispersion of Surface Drifters in the Benguela Upwelling Region." Journal of Physical Oceanography 48, no. 10 (October 2018): 2325–41. http://dx.doi.org/10.1175/jpo-d-18-0027.1.
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AbstractWe examine the relative dispersion of surface drifters deployed in groups of triplets at the boundaries of a filament in the upwelling region off Namibia for both the entire ensemble and the two main subgroups. For the drifters in the group released at the northern boundary of the filament, close to the upwelling front, we find that the mean-square pair separation 〈s2(t)〉 shows the characteristic distinct dispersion regimes [nonlocal, local (Richardson), and diffusive] of an ocean surface mixed layer. We confirm the different dispersion regimes by a rescaled presentation of the moments 〈sn(t)〉 and thereby also explain the anomalous slow decay of the kurtosis in the transient regime. For the drifter group released at the southern boundary, 〈s2(t)〉 remains constant for a short period, followed by a steep “Richardson like” increase and an asymptotic diffusive increase. In contrast to the northern release, the corresponding moments reveal a narrow distribution of pair separations for all regimes. The analysis of finite-size Lyapunov exponents (FSLEs) reveals consistent results when applied to the two releases separately. When applied to the entire drifter ensemble, the two measures yield inconsistent results. We relate the breakdown of consistency to the impact of the different dynamics on the respective averages: whereas, because of separation in scale, 〈s2(t)〉 is dominated by the northern release, the decay of the FSLEs for small distances reflects the drifter dynamics within the filament.
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Hernández-Carrasco, Ismael, Lohitzune Solabarrieta, Anna Rubio, Ganix Esnaola, Emma Reyes, and Alejandro Orfila. "Impact of HF radar current gap-filling methodologies on the Lagrangian assessment of coastal dynamics." Ocean Science 14, no. 4 (August 24, 2018): 827–47. http://dx.doi.org/10.5194/os-14-827-2018.
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Abstract. High-frequency radar, HFR, is a cost-effective monitoring technique that allows us to obtain high-resolution continuous surface currents, providing new insights for understanding small-scale transport processes in the coastal ocean. In the last years, the use of Lagrangian metrics to study mixing and transport properties has been growing in importance. A common condition among all the Lagrangian techniques is that complete spatial and temporal velocity data are required to compute trajectories of virtual particles in the flow. However, hardware or software failures in the HFR system can compromise the availability of data, resulting in incomplete spatial coverage fields or periods without data. In this regard, several methods have been widely used to fill spatiotemporal gaps in HFR measurements. Despite the growing relevance of these systems there are still many open questions concerning the reliability of gap-filling methods for the Lagrangian assessment of coastal ocean dynamics. In this paper, we first develop a new methodology to reconstruct HFR velocity fields based on self-organizing maps (SOMs). Then, a comparative analysis of this method with other available gap-filling techniques is performed, i.e., open-boundary modal analysis (OMA) and data interpolating empirical orthogonal functions (DINEOFs). The performance of each approach is quantified in the Lagrangian frame through the computation of finite-size Lyapunov exponents, Lagrangian coherent structures and residence times. We determine the limit of applicability of each method regarding four experiments based on the typical temporal and spatial gap distributions observed in HFR systems unveiled by a K-means clustering analysis. Our results show that even when a large number of data are missing, the Lagrangian diagnoses still give an accurate description of oceanic transport properties.
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Han, Xiaolei, Jiawei Li, Hiroka Rinoshika, Yuyang Zhou, Yan Zheng, Lin Dong, and Akira Rinoshika. "Dynamics of multi-scale vortical structures behind a barchan dune." Physics of Fluids, February 18, 2023. http://dx.doi.org/10.1063/5.0131631.
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In this study, multi-scale vortical structures and vortex dynamics around a fixed-bed barchan dune have been experimentally investigated based on the particle image velocimetry (PIV) technique, wavelet transform, and the finite-time Lyapunov exponent (FTLE) method. It was found that the dynamic characteristics of a dune wake are predominated by large- and intermediate-scale coherent structures. Quadrant analysis of the Reynolds-stress distribution for the corresponding wavelet components revealed that ejection and sweep events are the main contributors to the whole field, while outward and inward interaction events just dominate the region near the dune crest. In addition, the process of ejection and sweeping motions associated with the turbulent bursting sequence can also be captured by applying proper orthogonal decomposition (POD) analysis of the decomposed velocity filed for the different wavelet components. Finally, a continuous development process of the different wavelet scale structures in the shear layer was visualized in the Lagrangian framework. The small-scale waves grow exponentially and gradually develop into larger-scale vortices when convected downstream until the reattachment point, and larger-scale vortices break into the smaller ones.
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LaCasce, J. H., and Thomas Meunier. "Relative dispersion with finite inertial ranges." Journal of Fluid Mechanics 932 (December 9, 2021). http://dx.doi.org/10.1017/jfm.2021.1050.
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Relative dispersion experiments are often analysed using theoretical predictions from two- and three-dimensional turbulence. These apply to infinite inertial ranges, assuming the same dispersive behaviour over all scales. With finite inertial ranges, the metrics are less conclusive. We examine this using pair separation probability density functions (PDFs), obtained by integrating a Fokker–Planck equation with different diffusivity profiles. We consider time-based metrics, such as the relative dispersion, and separation-based metrics, such as the finite scale Lyapunov exponent (FSLE). As the latter cannot be calculated from a PDF, we introduce a new measure, the cumulative inverse separation time (CIST), which can. This behaves like the FSLE, but advantageously has analytical solutions in the inertial ranges. This allows the establishment of consistency between the time- and space-based metrics, something which has been lacking previously. We focus on three dispersion regimes: non-local spreading (as in a two-dimensional enstrophy inertial range), Richardson dispersion (as in an energy inertial range) and diffusion (for uncorrelated pair motion). The time-based metrics are more successful with non-local dispersion, as the corresponding PDF applies from the initial time. Richardson dispersion is barely observed, because the self-similar PDF applies only asymptotically in time. In contrast, the separation-based CIST correctly captures the dependencies, even with a short (one decade) inertial range, and is superior to the traditional FSLE at large scales. Nevertheless, it is advantageous to use all measures together, to seek consistent indications of the dispersion.
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Grosvenor, Kevin, and Ro Jefferson. "The edge of chaos: quantum field theory and deep neural networks." SciPost Physics 12, no. 3 (March 3, 2022). http://dx.doi.org/10.21468/scipostphys.12.3.081.
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We explicitly construct the quantum field theory corresponding to a general class of deep neural networks encompassing both recurrent and feedforward architectures. We first consider the mean-field theory (MFT) obtained as the leading saddlepoint in the action, and derive the condition for criticality via the largest Lyapunov exponent. We then compute the loop corrections to the correlation function in a perturbative expansion in the ratio of depth T to width N, and find a precise analogy with the well-studied O(N) vector model, in which the variance of the weight initializations plays the role of the 't Hooft coupling. In particular, we compute both the O(1) corrections quantifying fluctuations from typicality in the ensemble of networks, and the subleading O(T/N) corrections due to finite-width effects. These provide corrections to the correlation length that controls the depth to which information can propagate through the network, and thereby sets the scale at which such networks are trainable by gradient descent. Our analysis provides a first-principles approach to the rapidly emerging NN-QFT correspondence, and opens several interesting avenues to the study of criticality in deep neural networks.
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Perez, Gabriel M. P., Pier Luigi Vidale, Helen Dacre, and Jorge L. García-Franco. "Using a Synoptic-scale Mixing Diagnostic to Explain Global Precipitation Variability from Weekly to Interannual Timescales." Journal of Climate, September 6, 2022, 1–41. http://dx.doi.org/10.1175/jcli-d-22-0110.1.
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Abstract Precipitation often happens along organised filaments or bands of moisture such as convergence zones. Recent regional studies have shown that these moisture filaments arise from synoptic-scale mixing features known as attracting “Lagrangian Coherent Structures” (LCSs). In this study, we present a climatology of synoptic-scale mixing and investigate its co-variability with precipitation on temporal scales ranging from weekly to interannual. We characterise mixing with the Finite-time Lyapunov Exponent (FTLE), a measure of parcel deformation, in ERA5 reanalysis data between 1980 and 2009. Attracting LCSs are identified as ridges of the FTLE. At the interannual time scale, we compare El Niño and La Niña events and find that composites of precipitation and mixing anomalies share similar spatial patterns. We also compare summer and winter seasons and find that composites of seasonal-mean precipitation and mixing anomalies present similar characteristics; i.e., precipitation is particularly intense (weak) where mixing is strong (weak). In particular, these patterns closely match the typical signatures of the Intertropical Convergence Zone (ITCZ) and monsoon systems and the migrations of extratropical cyclone tracks. At the subseasonal scale, we employ daily composites to investigate the influence of the Madden-Julian Oscillation and the North Atlantic Oscillation on the mixing regimes of the Atlantic and East Pacific; our results indicate that these oscillations control the synoptic-scale horizontal mixing and the occurrence of LCSs as to suppress or enhance precipitating systems like the ITCZ and the South Atlantic Convergence Zone. The results presented in this first climatology of synoptic-scale mixing and LCSs indicate that these are powerful diagnostics to identify circulation mechanisms underlying precipitation variability.
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Cheng, Hang, Hao Jiang, Kai Leong Chong, Quan Zhou, Yulu Liu, and Zhiming Lu. "The effect of surface roughness on the Lagrangian coherent structures in turbulent Rayleigh-Bénard convection." Physics of Fluids, August 25, 2022. http://dx.doi.org/10.1063/5.0103755.
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We perform direct numerical simulations of turbulent Rayleigh-Bénard convection in a closed square cell with roughness conducting plates at Rayleigh number fixed at Ra = 108 and the Prandtl number fixed at Pr = 1. To gain insight into the effect of surface roughness on material transport in turbulent Rayleigh-Bénard convection, the Lagrangian coherent structures (LCSs) are extracted using the finite-time Lyapunov exponent (FTLE) method in the cases of different roughness height. Firstly, we find that lobe structures are widely present in RB convection and we elucidate how they play a part in transporting heat from coner-flow rolls to large-scale circulation. Then, we quantify the heat flux along the LCSs, which contributes to 80% of the total flux. This implies that the LCSs play an important role in heat transport regardless of the roughness height. Furthermore, two different mechanisms of heat transport in RB convection induced by roughness heights are explained in the Lagrangian perspective: the decrease of Nu number in the cases of h < hc is caused by the LCSs between the roughness elements which hinders the exchange of material between the fluid in the cavity and the bulk region; whereas, the increase of Nu number in the case of h > hc is produced by the enhanced mixing events of the convection that enhance the contribution of heat transport in the bulk region.
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Reijtenbagh, Jesse, Jerry Westerweel, and Willem Van de Water. "Large-scale structures of scalar and velocity in a turbulent jet flow." 14th International Symposium on Particle Image Velocimetry 1, no. 1 (August 1, 2021). http://dx.doi.org/10.18409/ispiv.v1i1.28.
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We study the relation between large-scale structures in the concentration field with those in the velocity field in a dye-seeded turbulent jet. The scalar concentration in a plane is measured using laser-induced fluorescence. Uniform concentration zones of an advected scalar are identified using cluster analysis. We simultaneously measure the two-dimensional velocity field using particle image velocimetry. The structures in the velocity field are characterized by finite-time Lyapunov exponents. The measurement of the scalarand velocity fields moves with the mean flow. In this moving frame, turbulent structures remain in focus long enough to observe well-defined ridges of the finite-time Lyapunov field. This field gauges the rate of point separation along Lagrangian trajectories; it was measured both for future and past times since the instant of observation. The edges of uniform concentration zones are correlated with the ridges of the past-time Lyapunov field, but not with those of the future-time Lyapunov field.
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De Leo, Annalisa, Francesco Enrile, and Alessandro Stocchino. "Periodic Lagrangian Coherent Structures around a tidal inlet." Frontiers in Marine Science 9 (August 18, 2022). http://dx.doi.org/10.3389/fmars.2022.959304.
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We present an extensive experimental campaign dedicated to the identification of coherent trajectory patterns owing to flow occurring in tidal environments, characterized by a tidal inlet and a channel with lateral tidal flats. Single and multiple harmonics tides are here reproduced on a large-scale physical model. The study of the large scale macro-vortices, generated by vortex shedding during the flood phase from the inlet barrier, is performed employing the Lagrangian Average Vorticity Deviation (LAVD). The presence of large-scale vortices with a complex dynamics within a tidal period suggested a deeper understanding on the possible environmental implications in terms of transport connections or barriers. Finite Time Lyapunov Exponents are employed in order to recognize stable and unstable manifolds within the flow that are defined as preferred paths along which particles are repelled (forward integration) or attracted (backward).
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Forgia, Giovanni La, Davide Cavaliere, Stefania Espa, Federico Falcini, and Guglielmo Lacorata. "Numerical and experimental analysis of Lagrangian dispersion in two-dimensional chaotic flows." Scientific Reports 12, no. 1 (May 6, 2022). http://dx.doi.org/10.1038/s41598-022-11350-1.
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AbstractWe present a review and a new assessment of the Lagrangian dispersion properties of a 2D model of chaotic advection and diffusion in a regular lattice of non stationary kinematic eddies. This model represents an ideal case for which it is possible to analyze the same system from three different perspectives: theory, modelling and experiments. At this regard, we examine absolute and relative Lagrangian dispersion for a kinematic flow, a hydrodynamic model (Delft3D), and a laboratory experiment, in terms of established dynamical system techniques, such as the measure of (Lagrangian) finite-scale Lyapunov exponents (FSLE). The new main results concern: (i) an experimental verification of the scale-dependent dispersion properties of the chaotic advection and diffusion model here considered; (ii) a qualitative and quantitative assessment of the hydro-dynamical Lagrangian simulations. The latter, even though obtained for an idealized open flow configuration, contributes to the overall validation of the computational features of the Delft3D model.
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Xia, Qiong, Changming Dong, Yijun He, Gaocong Li, and Jihai Dong. "Lagrangian study of several long-lived Agulhas rings." Journal of Physical Oceanography, February 17, 2022. http://dx.doi.org/10.1175/jpo-d-21-0079.1.
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Abstract By using a Lagrangian-Averaged Vorticity Deviation (LAVD)-based vortex detection scheme, rotationally coherent Lagrangian vortices in the South Atlantic Ocean are detected. These vortices act as agents for water transport and can stay coherent in a limited time scale. Our study starts from the life cycle of several long-lived Agulhas rings detected with the LAVD-based vortex detection method. The life cycle of those long-lived Agulhas rings can be separated into two distinct stages: the growing stage and the decaying stage. It is found that at the growing stage, the ambient water spins in and provides effective shielding for the coherent core. The rate of change of material belt width with respect to the detection time scale at the end of the growing stage can represent the decay rate of coherence. We further find a linear relationship between the mean strain rate and the mean square root of kinetic energy (KE½). Mean finite-time Lyapunov exponents (FTLE) increases monotonically with the mean strain rate or mean KE½. The long existence of the Agulhas rings can be partly attributed to the energetic boundaries around the rings. The ratio of the boundary kinetic energy to the spatial mean kinetic energy (KE/MKE) is also found to be a contributing factor that can influence the lifetime of Agulhas rings. In the retroflection area, the short-lived Agulhas rings might be attributed to the low KE/MKE in this area.