Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence

On the evidence of helico-spiralling recirculation within coherent cores of eddies using Lagrangian approach

Oceanic eddies exhibit remarkable coherence and longevity compared to other transient features in the surrounding flow. They possess the ability to transport properties over extensive distances while maintaining their material identity intact. The Lagrangian Coherent Structure (LCS) framework has proven effective in capturing these coherent eddies, where they display a solid-body-like rotation. Although various LCS approaches have been employed to investigate different facets of coherent eddies, a comprehensive understanding of their three-dimensional structures and internal dynamics remains elusive. This study aims to advance our comprehension of coherent eddies' structural characteristics and delve into the precise nature of their internal dynamics by utilizing the Lagrangian Averaged Vorticity Deviation approach. Two eddies, one cyclonic and the other anti-cyclonic, were chosen from a high-resolution simulation carried out in the Bay of Bengal using the Regional Ocean Modeling System (ROMS). The findings unveil that these eddies have three-dimensional coherent cores resembling gently tapered cones that are broader at the surface and gradually narrow towards the bottom. Intriguingly, the dynamically coherent core of these eddies exhibits simultaneous upwelling and downwelling while maintaining their volumes during advection due to persistent material coherence. The three-dimensional trajectories followed by the fluid parcels inside the coherent core are helical. Their two-dimensional horizontal projections show alternating spiral bands of upwelling and downwelling which are the manifestations of Vortex Rossby Waves. These observations lead to a conceptual framework of a three-dimensional helico-spiralling recirculation pattern within the coherent cores of eddies.

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.

Altimetry-Based Diagnosis of Deep-Reaching Sub-Mesoscale Ocean Fronts

Recent studies demonstrate that energetic sub-mesoscale fronts (10–50 km width) extend in the ocean interior, driving large vertical velocities and associated fluxes. However, diagnosing the dynamics of these deep-reaching fronts from in situ observations remains challenging because of the lack of information on the 3-D structure of the horizontal velocity. Here, a realistic numerical simulation in the Antarctic Circumpolar Current (ACC) is used to study the dynamics of submesocale fronts in relation to velocity gradients, responsible for the formation of these fronts. Results highlight that the stirring properties of the flow at depth, which are related to the velocity gradients, can be inferred from finite-size Lyapunov exponent (FSLE) at the surface. Satellite altimetry observations of FSLE and velocity gradients are then used in combination with recent in situ observations collected by an elephant seal in the ACC to reconstruct frontal dynamics and their associated vertical velocities down to 500 m. The approach proposed here is well suited for the analysis of sub-mesoscale-resolving datasets and the design of future sub-mesoscale field campaigns.

Critical transitions and perturbation growth directions

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

Reynolds number effects on mixing due to topological chaos

Topological chaos has emerged as a powerful tool to investigate fluid mixing. While this theory can guarantee a lower bound on the stretching rate of certain material lines, it does not indicate what fraction of the fluid actually participates in this minimally mandated mixing. Indeed, the area in which effective mixing takes place depends on physical parameters such as the Reynolds number. To help clarify this dependency, we numerically simulate the effects of a batch stirring device on a 2D incompressible Newtonian fluid in the laminar regime. In particular, we calculate the finite time Lyapunov exponent (FTLE) field for three different stirring protocols, one topologically complex (pseudo-Anosov) and two simple (finite-order), over a range of viscosities. After extracting appropriate measures indicative of both the amount of mixing and the area of effective mixing from the FTLE field, we see a clearly defined Reynolds number range in which the relative efficacy of the pseudo-Anosov protocol over the finite-order protocols justifies the application of topological chaos. More unexpectedly, we see that while the measures of effective mixing area increase with increasing Reynolds number for the finite-order protocols, they actually exhibit non-monotonic behavior for the pseudo-Anosov protocol.

Monitoring cooling water discharge using Lagrangian coherent structures: a case study in Daya Bay, China

As an unwanted by-product in the power plants, cooling water (CW) discharge may induce harmful effects on the coastal environment. In this paper, to accurately predict the CW transport in the Daya Bay, China, we develop a three-dimensional hydrodynamic and temperature model to simulate the transport of the CW from nuclear power plant into coastal waters. Specifically, we use the flow velocity produced by the model to locate Lagrangian coherent structures (LCSs) hidden in ocean surface currents. Result show that the transport of the CW is quite strongly tied to the simulated LCSs. The LCSs constitute fluid barriers that accurately demarcate potential pathway for CW transport. Thus, LCSs in velocity fields from a coastal model is an extremely useful way to monitor and interpret the transport of CW.

Fast chemical reaction in two-dimensional Navier-Stokes flow: initial regime

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion κ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent λ with two times τ and τ[over ̃]. The time 1/λ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while τ measures it in the recent past before t, and τ[over ̃] in the early part of the trajectory. We show that the chemical speed scales like κ(1/2) and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.

Barriers to transport induced by periodic oscillations in a physical model of the human vitreous chamber

Understanding mixing processes that occur in the human vitreous chamber is of fundamental importance due to the relevant clinical implications in drug delivery treatments of several eye conditions. In this article we rely on experimental observations (which demonstrated that dispersion coefficients largely dominate diffusive coefficients) on a physical model of the human eye to perform an analysis based on Lagrangian trajectories. In particular, we study barriers to transport in a particularly significant two-dimensional section of the eye model by using nonlinear dynamical systems theoretical and numerical tools. Bifurcations in the system dynamics are investigated by varying the main physical parameters of the problem.

Using hyperbolic Lagrangian coherent structures to investigate vortices in bioinspired fluid flows

We use direct Lyapunov exponents to identify Lagrangian coherent structures (LCSs) in a bioinspired fluid flow: the wakes of rigid pitching panels with a trapezoidal planform geometry chosen to model idealized fish caudal fins. When compared with commonly used Eulerian criteria, the Lagrangian method has previously exhibited the ability to define structure boundaries without relying on a preselected threshold. In addition, qualitative changes in the LCS have previously been shown to correspond to physical changes in the vortex structure. For this paper, digital particle image velocimetry experiments were performed to obtain the time-resolved velocity fields for Strouhal numbers of 0.17 and 0.27. A classic reverse von Karman vortex street pattern was observed along the midspan of the near wake at low Strouhal number, but at higher Strouhal number the complexity of the wake increased downstream of the trailing edge. The spanwise vortices spread transversely across the wake and lose coherence, and this event was shown to correspond to a qualitative change in the LCS at the same time and location.

Smooth and filamental structures in chaotically advected chemical fields

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large Péclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, gamma(q), of the qth -order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, alpha , and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), h. Under a homogeneous stretching approximation, gamma(q)/q=max[alpha/h,1] which implies that a well-defined filamental-smooth transition occurs at alpha=h. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as q varies and less on the effects of alpha and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, [symbol: see text] depend nonlinearly on alpha with [formula: see text]. The magnitude of the correction to the homogeneous stretching approximation, [formula: see text], grows as alpha increases, reaching a maximum when the leading-order transition is reached (alpha=h). The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for alpha>h(max), where h(max) denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for alpha<h, with an intermediate character for alpha between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Crámer function.

Exponential localization of singular vectors in spatiotemporal chaos

In a dynamical system the singular vector (SV) indicates which perturbation will exhibit maximal growth after a time interval tau . We show that in systems with spatiotemporal chaos the SV exponentially localizes in space. Under a suitable transformation, the SV can be described in terms of the Kardar-Parisi-Zhang equation with periodic noise. A scaling argument allows us to deduce a universal power law tau(-gamma) for the localization of the SV. Moreover the same exponent gamma characterizes the finite- tau deviation of the Lyapunov exponent in excellent agreement with simulations. Our results may help improve existing forecasting techniques.

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Some statistical properties of finite-time stability exponents in the standard map can be estimated analytically. The mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants. Whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations, the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength. In particular, the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings. The average of the stability matrix' inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter. The usual accelerator modes and the small islands created through double saddle node bifurcations, which come halfway between the positions in interaction strength of the usual accelerator modes, are clearly visible in the variance, whose time scaling is capable of detecting the presence of islands as small as 0.01% of the phase space. We study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions. We also discuss the nature of this local approximation in some detail.

Mixing efficiency in an excitable medium with chaotic shear flow

The effect of a time-periodic chaotic shear flow on an excitable chemical medium is studied numerically. Stirring effects on pattern formation strongly depend on the shear amplitude and the ratio of the advective and chemical time scales (Damköhler number, Da). We have observed that the wave period increases with decreasing Da below some critical value, afterwards the period decreases until complete wave annihilation. In the last case, before final extinction, a set of uncorrelated, nonstationary excitable dots survive, whose number depends on the mixing rate. Insights on the nature of this critical behavior are obtained through the calculation of the mixing efficiency of the flow.

Foundations of chaotic mixing

The simplest mixing problem corresponds to the mixing of a fluid with itself; this case provides a foundation on which the subject rests. The objective here is to study mixing independently of the mechanisms used to create the motion and review elements of theory focusing mostly on mathematical foundations and minimal models. The flows under consideration will be of two types: two-dimensional (2D) 'blinking flows', or three-dimensional (3D) duct flows. Given that mixing in continuous 3D duct flows depends critically on cross-sectional mixing, and that many microfluidic applications involve continuous flows, we focus on the essential aspects of mixing in 2D flows, as they provide a foundation from which to base our understanding of more complex cases. The baker's transformation is taken as the centrepiece for describing the dynamical systems framework. In particular, a hierarchy of characterizations of mixing exist, Bernoulli --> mixing --> ergodic, ordered according to the quality of mixing (the strongest first). Most importantly for the design process, we show how the so-called linked twist maps function as a minimal picture of mixing, provide a mathematical structure for understanding the type of 2D flows that arise in many micromixers already built, and give conditions guaranteeing the best quality mixing. Extensions of these concepts lead to first-principle-based designs without resorting to lengthy computations.

Stretching, alignment, and shear in slowly varying velocity fields

We derive criteria that locate intense material stretching and shear in two-dimensional flows with slow time dependence. Our derivation makes use of the near integrability of the equation of variations along trajectories of the slowly varying flow. The criteria yield two diagnostic scalar fields for use in real-time Lagrangian predictions in geophysical flows.