Nonasymptotic properties of transport and mixing

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.

Effects of large-scale advection and small-scale turbulent diffusion on vertical phytoplankton dynamics

Turbulence has been recognized as a factor of paramount importance for the survival or extinction of sinking phytoplankton species. However, dealing with its multiscale nature in models of coupled fluid and biological dynamics is a formidable challenge. Advection by coherent structures, such as those related to winter convection and Langmuir circulation, is also recognized to play a role in the survival and localization of phytoplankton. In this work we revisit a theoretically appealing model for phytoplankton vertical dynamics, and numerically investigate how large-scale fluid motions affect the survival conditions and the spatial distribution of the biological population. For this purpose, and to work with realistic parameter values, we adopt a kinematic flow field to account for the different spatial and temporal scales of turbulent motions. The dynamics of the population density are described by an advection-reaction-diffusion model with a spatially heterogeneous growth term proportional to sunlight availability. We explore the role of fluid transport by progressively increasing the complexity of the flow in terms of spatial and temporal scales. We find that, due to the large-scale circulation, phytoplankton accumulates in downwelling regions and its growth is reduced, confirming previous indications in slightly different conditions. We then explain the observed phenomenology in terms of a plankton filament model. Moreover, by contrasting the results in our different flow cases, we show that the large-scale coherent structures have an overwhelming importance. Indeed, we find that smaller-scale motions only quite weakly affect the dynamics, without altering the general mechanism identified. Such results are relevant for parametrizations in numerical models of phytoplankton life cycles in realistic oceanic flow conditions.

The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

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

Seascape connectivity of European anchovy in the Central Mediterranean Sea revealed by weighted Lagrangian backtracking and bio-energetic modelling

Ecological connectivity is one of the most important processes that shape marine populations and ecosystems, determining their distribution, persistence, and productivity. Here we use the synergy of Lagrangian back-trajectories, otolith-derived ages of larvae, and satellite-based chlorophyll-a to identify spawning areas of European anchovy from ichthyoplanktonic data, collected in the Strait of Sicily (Central Mediterranean Sea), i.e., the crucial channel in between the European and African continents. We obtain new evidence of ecosystem connectivity between North Africa and recruitment regions off the southern European coasts. We assess this result by using bio-energetic modeling, which predicts species-specific responses to environmental changes by producing quantitative information on functional traits. Our work gives support to a collaborative and harmonized use of Geographical Sub-Areas, currently identified by the General Fisheries Commission for the Mediterranean. It also confirms the need to incorporate climate and environmental variability effects into future marine resources management plans, strategies, and directives.

Effect of small scale transport processes on phytoplankton distribution in coastal seas

Coastal ocean ecosystems are major contributors to the global biogeochemical cycles and biological productivity. Physical factors induced by the turbulent flow play a crucial role in regulating marine ecosystems. However, while large-scale open-ocean dynamics is well described by geostrophy, the role of multiscale transport processes in coastal regions is still poorly understood due to the lack of continuous high-resolution observations. Here, the influence of small-scale dynamics (O(3.5–25) km, i.e. spanning upper submesoscale and mesoscale processes) on surface phytoplankton derived from satellite chlorophyll-a (Chl-a) is studied using Lagrangian metrics computed from High-Frequency Radar currents. The combination of complementary Lagrangian diagnostics, including the Lagrangian divergence along fluid trajectories, provides an improved description of the 3D flow geometry which facilitates the interpretation of two non-exclusive physical mechanisms affecting phytoplankton dynamics and patchiness. Attracting small-scale fronts, unveiled by backwards Lagrangian Coherent Structures, are associated to negative divergence where particles and Chl-a standing stocks cluster. Filaments of positive divergence, representing large accumulated upward vertical velocities and suggesting accrued injection of subsurface nutrients, match areas with large Chl-a concentrations. Our findings demonstrate that an accurate characterization of small-scale transport processes is necessary to comprehend bio-physical interactions in coastal seas.

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.

Asymptotic diffusion coefficients and anomalous diffusion in a meandering jet flow under environmental fluctuations

The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which is linearly dependent to the flow's meander amplitude. Increasing the noise the fluid transport passes through a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner circulation regimes in the flow has been found to be determinant to explain the observed behavior.

Collective motion of self-propelled particles interacting without cohesion

We present a comprehensive study of Vicsek-style self-propelled particle models in two and three space dimensions. The onset of collective motion in such stochastic models with only local alignment interactions is studied in detail and shown to be discontinuous (first-order-like). The properties of the ordered, collectively moving phase are investigated. In a large domain of parameter space including the transition region, well-defined high-density and high-order propagating solitary structures are shown to dominate the dynamics. Far enough from the transition region, on the other hand, these objects are not present. A statistically homogeneous ordered phase is then observed, which is characterized by anomalously strong density fluctuations, superdiffusion, and strong intermittency.

Finite-size Lyapunov exponent for Levy processes

The finite-size Lyapunov exponent (FSLE) is the exponential rate at which two particles separate from a distance of r to a x r (a>1) and provides a measure of dispersive mixing in chaotic systems. It is shown analytically that for particle trajectories governed by symmetric alpha -stable Levy motion, the FSLE is proportional to the diffusion coefficient and inversely proportional to r(alpha). This power law provides an easy method to determine the parameters for Levy processes and hence has applications to superdiffusion in the atmospheric, oceanic, and terrestrial sciences.

Application of the finite-size Lyapunov exponent to particle tracking velocimetry in fluid mechanics experiments

A finite-size (or scale) Lyapunov exponent (FSLE), lambdaa(x), is presented in a statistical mechanical framework and employed to characterize mixing in a variety of laboratory and computational fluid mechanics experiments. The FSLE is the exponential rate at which two particles separate from a distance x to ax. Laboratory particle tracking experiments are used to study penetrative convection and flow in porous media while computational experiments are used to study Lévy processes and deterministic diffusion. The apparent scaling relation lambaa(x) approximately Cax(-beta(a)) of the FSLE holds over intermediate initial separations where the laboratory experiment data is most accurate and asymptotically for the computational experiments. The dependence of the exponent beta on a decreases with increasing a. In the matched index porous system, Ca is also a function of mean fluid velocity. The exponent beta is alpha when the Lévy process is alpha-stable and in this case beta is independent of a.

Mixing and reaction efficiency in closed domains

We present a numerical study of mixing and reaction efficiency in closed domains. In particular, we focus our attention on laminar flows. In the case of inert transport the mixing properties of the flows strongly depend on the details of the Lagrangian transport. We also study the reaction efficiency. Starting with a little spot of product, we compute the time needed to complete the reaction in the container. We find that the reaction efficiency is not strictly related to the mixing properties of the flow. In particular, reaction acts as a "dynamical regulator".

Hyperbolic lines and the stratospheric polar vortex

The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for Medium-Range Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another (still nonrigorous) alternative is the finite-size Lyapunov exponent (FSLE) which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures (ECS) in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECS-like barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows. (c) 2002 American Institute of Physics.

Front speed enhancement in cellular flows

The problem of front propagation in a stirred medium is addressed in the case of cellular flows in three different regimes: slow reaction, fast reaction and geometrical optics limit. It is well known that a consequence of stirring is the enhancement of front speed with respect to the nonstirred case. By means of numerical simulations and theoretical arguments we describe the behavior of front speed as a function of the stirring intensity, U. For slow reaction, the front propagates with a speed proportional to U(1/4), conversely for fast reaction the front speed is proportional to U(3/4). In the geometrical optics limit, the front speed asymptotically behaves as U/ln U. (c) 2002 American Institute of Physics.

Lagrangian chaos and Eulerian chaos in shear flow dynamics

Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.

Advection of passive particles in the Kolmogorov flow

A statistical analysis of the advection of passive particles in a flow governed by driven two-dimensional Navier-Stokes equations (Kolmogorov flow) is presented. Different regimes are studied, all corresponding to a chaotic behavior of the flow. The diffusion is found to be strongly asymmetric with a very weak transport perpendicular to the forcing direction. The trajectories of the particles are characterized by the presence of traps and flights. The trapping time distributions show algebraic decrease, and strong anomalous diffusion is observed in transient phases. Different regimes lead to different types of diffusion, i.e., no universal behavior of diffusion is observed, and both time and space properties are needed to define anomalous transport. (c) 2001 American Institute of Physics.