Finite-scale Lyapunov Exponent Research Articles

Anisotropic Lagrangian dispersion in zonostrophic turbulence in a closed basin

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.

Numerical and experimental analysis of Lagrangian dispersion in two-dimensional chaotic flows.

We 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.

Relative dispersion with finite inertial ranges

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.

Relative Dispersion of Surface Drifters in the North Sea: The Effect of Tides on Mesoscale Diffusivity

AbstractWe examine the relative dispersion and the contribution of tides on the relative diffusivities of surface drifters in the North Sea. The drifters are released in two clusters, yielding 43 pairs, in the vicinity of a tidal mixing front in the German Bight, which is located in the southeastern area of the North Sea. Both clusters indicate decreasing dispersion when crossing the tidal mixing front, followed by exponentially increasing dispersion with e‐folding times of 0.5 days for Cluster 1 and 0.3 days for Cluster 2. A transition of the dispersion regimes is observed at scales of the order of the Rossby radius of deformation (10 km). After that, the relative dispersion grows with a power‐law dependency with a short period of ballistic dispersion (quadratic growth), followed by a Richardson regime (cubic growth) in the final phase. Scale‐dependent metrics such as the relative diffusivities are consistent with these findings, while the analysis of the finite‐scale Lyapunov exponents (FSLEs) shows contradictory results for the submesoscales. In summary, the analysis of various statistical Lagrangian metrics suggests that tracer stirring at the submesoscales is nonlocal and becomes local at separation scales larger than 10 km. The analysis of meridional and zonal dispersion components indicates anisotropic dispersion at the submesoscales, which changes into isotropic dispersion on the mesoscales. Spectral analysis of the relative diffusivity gives evidence that semidiurnal and shallow‐water tides influence relative diffusivity at the mesoscales, especially for drifter separations above 50 km.

FSLE analysis and validation of Lagrangian simulations based on satellite-derived GlobCurrent velocity data

We present an independent validation of the satellite surface velocity fields released by the ESA Data User Element GlobCurrent Project. The validation methodology is based on the analysis of Lagrangian numerical trajectories integrated from satellite-based sea surface currents. Two case studies were considered: Mediterranean Sea and North Atlantic Ocean.The Finite-Scale Lyapunov Exponents provide a rigorous, quantitative tool to evaluate Lagrangian simulations with respect to real drifter trajectories. An accuracy threshold scale can be identified as the scale above which the error size, propagating along a numerical trajectory, grows no faster than the relative separation between real drifters. Below this threshold, the error growth rate tends to diverge linearly as the error decreases. The mean error growth speed, at early stage, is found to be related to the kinetic energy of the missing scales of motion, not resolved by the GlobCurrent products. Established kinematic Lagrangian models are, also, exploited to compensate the energy gap between real and numerical trajectories in the unresolved scale range. Ultimately, GlobCurrent surface velocity fields are shown to have overall good ‘Lagrangian skills' for large-scale transport and dispersion numerical simulations.

Chaotic Lagrangian models for turbulent relative dispersion.

A deterministic multiscale dynamical system is introduced and discussed as a prototype model for relative dispersion in stationary, homogeneous, and isotropic turbulence. Unlike stochastic diffusion models, here trajectory transport and mixing properties are entirely controlled by Lagrangian chaos. The anomalous "sweeping effect," a known drawback common to kinematic simulations, is removed through the use of quasi-Lagrangian coordinates. Lagrangian dispersion statistics of the model are accurately analyzed by computing the finite-scale Lyapunov exponent (FSLE), which is the optimal measure of the scaling properties of dispersion. FSLE scaling exponents provide a severe test to decide whether model simulations are in agreement with theoretical expectations and/or observation. The results of our numerical experiments cover a wide range of "Reynolds numbers" and show that chaotic deterministic flows can be very efficient, and numerically low-cost, models of turbulent trajectories in stationary, homogeneous, and isotropic conditions. The mathematics of the model is relatively simple, and, in a geophysical context, potential applications may regard small-scale parametrization issues in general circulation models, mixed layer, and/or boundary layer turbulence models as well as Lagrangian predictability studies.

General characteristics of relative dispersion in the ocean

The multi-scale and nonlinear nature of the ocean dynamics dramatically affects the spreading of matter, like pollutants, marine litter, etc., of physical and chemical seawater properties, and the biological connectivity inside and among different basins. Based on the Finite-Scale Lyapunov Exponent analysis of the largest available near-surface Lagrangian data set from the Global Drifter Program, our results show that, despite the large variety of flow features, relative dispersion can ultimately be described by a few parameters common to all ocean sub-basins, at least in terms of order of magnitude. This provides valuable information to undertake Lagrangian dispersion studies by means of models and/or of observational data. Moreover, our results show that the relative dispersion rates measured at submesoscale are significantly higher than for large-scale dynamics. Auxiliary analysis of high resolution GPS-tracked drifter hourly data as well as of the drogued/undrogued status of the buoys is provided in support of our conclusions. A possible application of our study, concerning reverse drifter motion and error growth analysis, is proposed relatively to the case of the missing Malaysia Airlines MH370 aircraft.

Surface Relative Dispersion in the Southwestern Gulf of Mexico

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.

Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans.

Turbulence with inverse energy cascade and its transport properties are investigated experimentally in a flow associated with a westward propagating jet. Turbulence and the jet were produced by an electromagnetic force in a rotating tank filled with an electrolytic saline solution. The parabolic free surface emulated the topographic β effect which evoked the zonation. The spectral and transport flow characteristics were highly anisotropic. Turbulence is diagnosed by exploring the analogy between vertical and horizontal turbulent overturns in, respectively, stably stratified and quasigeostrophic flows which gives rise to a method of potential vorticity (PV) monotonizing. The anisotropization of transport properties of the flow is investigated using the finite scale Lyapunov exponent technique. After initial exponential particle separation, radial (meridional in geophysical and planetary applications) diffusion attains a short-ranged Richardson regime which transitions to the Taylor (scale-independent diffusivity) one. The azimuthal (zonal) diffusion exhibits a double-plateau structure which attains a superdiffusive regime on large scales. The transition to the Taylor regime for the radial diffusion takes place at a scale of turbulence anisotropization. The radial eddy diffusivity in both regimes as well as the transition scale are all determined by the rate of the inverse energy cascade, ε, that can be diagnosed by the PV monotonizing. Conversely, ε can be deduced from the scale of the Richardson-Taylor regime transition in the radial eddy diffusivity which, thus, provides an additional tool of diagnosing anisotropic macroturbulence with inverse energy cascade.

Material dispersion by oceanic internal waves

Internal gravity waves that are generated in the open ocean have a universal frequency spectrum, called Garrett–Munk spectrum. By initializing internal waves that satisfy the Garrett–Munk spectrum in a non-hydrostatic numerical model, we investigate the material dispersion produced by these internal waves. Three numerical experiments are designed: Exp.-1 uses a linearly stratified fluid, Exp.-2 has an upper mixed layer, and Exp.-3 incorporates a circular front into the upper mixed layer. Resorting to neutrally buoyant particles, we investigate the dispersion in terms of metrics of the relative dispersion and finite-scale Lyapunov exponent (FSLE). Exp.-1 shows that the dispersion regime produced by these internal waves is between ballistic and diffusive based on relative dispersion, and is however ballistic according to FSLE. The maximum FSLE at scales of 100 m is about 5 day\(^{-1}\), which is comparable to that calculated using ocean drifters. Exp.-2 demonstrates that internal waves can generate flows and material dispersion in an upper mixed layer. However, when mixed layer eddies are present, as in Exp.-3, the dispersion in the mixed layer is controlled by the eddies. In addition, we show that inertial oscillations do not affect the relative dispersion, but impact FSLE at scales of inertial oscillations.

Pairwise surface drifter separation in the western Pacific sector of the Southern Ocean

AbstractThe Southern Ocean plays a critical role in global climate, yet the mixing properties of the circulation in this part of the ocean remain poorly understood. Here dispersion in the vicinity of the Southern Antarctic Circumpolar Current Front, one of the branches of the Antarctic Circumpolar Current, is studied using 10 pairs of surface drifters deployed systematically across the frontal jet and its flanks. Drifter pairs were deployed with an initial separation of 13 m and report their position every hour. The separation of the pairs over 7 months, in terms of their Finite‐Scale Lyapunov Exponents (FSLE), dispersion, and diffusivity, is characterized and related to expected behavior from Quasi‐Geostrophic (QG) and Surface Quasi‐Geostrophic (SQG) theories. The FSLE analysis reveals two submesoscale regimes, with SQG‐like behavior at scales below 3.2 km and mixed QG/SQG behavior at scales between 3.2 and 73 km. The dispersion analysis, however, suggests QG‐like behavior for the smallest scales. Both dispersion and diffusivity appear isotropic for scales up to 500 km. Finally, there is no clear indication of a cross‐jet variation of drifter dispersion.

Surface dispersion in the Gulf of California

Surface dispersion is measured in the Gulf of California by means of Argos drifters released along this semi-enclosed, elongated basin. First, basic one-particle statistics (Lagrangian scales, absolute dispersion and diffusion coefficients) are estimated along and across the Gulf. Absolute dispersion shows a nearly ballistic regime during the Lagrangian time scale (<2days) in both directions (it grows as ∼t2, where t is time). During the subsequent 30days, absolute dispersion enters a random-walk regime (∼t) along the Gulf, while being saturated across the basin due to the lateral boundaries. Secondly, the analysis is extended to two-particle statistics (relative dispersion between pairs of drifters and Finite Scale Lyapunov Exponents, FSLE). Relative dispersion is nearly exponential in both directions during the first few days, though evidence is not conclusive. During the subsequent 30days, it grows as ∼t1.5 along the Gulf, while being saturated across the basin again. It is shown that relative dispersion along the Gulf is proportional to t̂3, where t̂ represents a shifted time that depends on the initial separation of the particles. This form of the Richardson regime is consistently measured for particles that are sufficiently separated (30km or more). The Richardson regime is verified with the FSLE for particle separations ranging from 30 to 140km, approximately. The obtained dispersion properties are discussed in terms of the main circulation features within the basin, such as mesoscale vortices that occupy the width of the Gulf. These structures might retain buoys during days or weeks, thus preventing or delaying further displacements and therefore affecting the particle dispersion. The vortices are also an important mechanism to translate particles across the Gulf, between the Peninsula and the continent, thus promoting the saturation of dispersion along this direction.

How Does Drifter Position Uncertainty Affect Ocean Dispersion Estimates?

AbstractTo 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 &lt; 3 km. An expression is provided to estimate position errors that can be afforded depending on the expected maximum λ in the submesoscale regime.

Lagrangian predictability characteristics of an Ocean Model

The Mediterranean Forecasting System (MFS) Ocean Model, provided by INGV, has been chosen as case study to analyze Lagrangian trajectory predictability by means of a dynamical systems approach. To this regard, numerical trajectories are tested against a large amount of Mediterranean drifter data, used as sample of the actual tracer dynamics across the sea. The separation rate of a trajectory pair is measured by computing the Finite-Scale Lyapunov Exponent (FSLE) of first and second kind. An additional kinematic Lagrangian model (KLM), suitably treated to avoid “sweeping”-related problems, has been nested into the MFS in order to recover, in a statistical sense, the velocity field contributions to pair particle dispersion, at mesoscale level, smoothed out by finite resolution effects. Some of the results emerging from this work are: (a) drifter pair dispersion displays Richardson's turbulent diffusion inside the [10–100] km range, while numerical simulations of MFS alone (i.e., without subgrid model) indicate exponential separation; (b) adding the subgrid model, model pair dispersion gets very close to observed data, indicating that KLM is effective in filling the energy “mesoscale gap” present in MFS velocity fields; (c) there exists a threshold size beyond which pair dispersion becomes weakly sensitive to the difference between model and “real” dynamics; (d) the whole methodology here presented can be used to quantify model errors and validate numerical current fields, as far as forecasts of Lagrangian dispersion are concerned.

Anisotropic Lagrangian Dispersion in Rotating Flows with a β Effect

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.

On the stirring properties of the thermally-driven rotating annulus

The stirring properties of the thermally-driven rotating annulus have not been extensively studied, despite sustained interest in the stirring properties of various geophysical flows, and the wide applicability of the rotating annulus to geophysical problems. This paper takes important steps towards a thorough investigation of the stirring properties of thermally-driven rotating annulus flows, by demonstrating numerically the utility of two stirring measures for a parameter set yielding relatively simple flow conditions. The first measure is the finite scale Lyapunov exponent (FSLE), which has been successfully used to highlight the stirring properties of various geophysical flows. The second measure is the Eulerian symmetry measure, which has been far less widely used: this second measure does not provide such a detailed view of the stirring properties as the FSLE, but is far more efficient to calculate. Both measures are shown to have some success for the simple flow case studied, providing a strong foundation for further investigation into more complicated flows.

On multi-scale dispersion under the influence of surface mixed layer instabilities and deep flows

A series of large eddy simulations is used to assess the transport properties of multi-scale ocean flows. In particular, we compare scale-dependent measures of Lagrangian relative dispersion and the evolution of passive tracer releases in models containing only submesoscale mixed layer instabilities and those containing mixed layer instabilities modified by deeper, baroclinic mesoscale disturbances. Visualization through 3D finite-time Lyapunov exponents and spectral analysis show that the small scale instabilities of the mixed layer rapidly lose coherence in the presence of larger-scale straining induced by the mesoscale motion. Eddy diffusivities computed from passive tracer evolution increase by an order of magnitude as the flow transitions from small to large scales. During the time period when both instabilities are present, scale-dependent relative Lagrangian dispersion, given by the finite-scale Lyapunov exponent (λ), shows two distinct plateau regions clearly associated with the disparate instability scales. In this case, the maximum value of λ over the submesocales at the surface flow is three times greater than λ at the mixed layer base which is only influenced by the deeper baroclinic motions. The results imply that parameterizations of submesoscale transport properties may be needed to accurately predict surface dispersion in models that do not explicitly resolve submesoscale turbulent features.

On the influence of a β‐effect on Lagrangian diffusion

Geophysical turbulence is strongly influenced by the variation of the Coriolis force with latitude (β‐effect): when this effect is significant an anisotropic inverse cascade is developed since energy is preferentially transferred towards zonal modes. The consequent emergence of flow structures along the zonal direction can strongly impact turbulent transport and modify meridional scalar diffusion. In this letter we investigate zonal and meridional diffusion in laboratory experiments of turbulence affected by aβ‐effect. The degree of anisotropy and the flow characteristic scales have been quantified according to a Lagrangian approach based on the reconstruction of tracer trajectories, i.e., in analogy with oceanic drifters data, and on the Finite‐Scale Lyapunov Exponent (FSLE) analysis. An experimental confirmation of a recently introduced scaling law for the meridional diffusion in the marginally zonostrophic regime is also presented.

Lagrangian Drifter Dispersion in the Southwestern Atlantic Ocean

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.

Lagrangian mixing in straight compound channels

Recently Stocchino &amp; 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.