Search

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude-frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred-thousand degrees of freedom., Nonlinear Dynamics, 107 (2), ISSN:0924-090X, ISSN:1573-269X

Lagrangian ocean drifters provide highly accurate approximations of ocean surface currents but are sparsely located across the globe. As drifters passively follow ocean currents, there is minimal control on where they will be making measurements, providing limited temporal coverage for a given region. Complementary Eulerian velocity data are available with global coverage but are themselveslimited by the spatial and temporal resolution possible with satellite altimetry measurements. In addition, altimetry measurements approximate geostrophic components of ocean currents but neglect smaller submesoscale motions and require smoothing and interpolation from raw satellite track measurements. In an effort to harness the rich dynamics available in ocean drifter datasets, we have trained a recurrent neural network on the time history of drifter motion to minimize the error in a reduced-order Maxey–Riley drifter model. This approach relies on a slow-manifold approximation to determine the most mathematically relevant variables with which to train, subsequently improving the temporal and spatial resolution of the underlying velocity field. By adding this neural-network component, we also correct drifter trajectories near submesoscale features missed by deterministic models using only satellite and wind reanalysis data. The effect of varying similarity between training and testing trajectory datasets for the blended model was evaluated, as was the effect of seasonality in the Gulf of Mexico.

Lagrangian coherent structures (LCS) provide a means to understand persistent flow features in an objective manner. There has been great success identifying and harnessing hyperbolic, elliptic, and parabolic structures in both oceanic and atmospheric flows. These approaches (e.g. FTLE, PRA, LAVD) rely on well resolved velocity information for the computation of the gradient of the flow map or vorticity deviation. Thus, for sparse data, such as that available from ocean drifters or atmospheric balloons, the quality of these methods quickly deteriorates. On the other hand, all elementary features of individual particle paths, such as velocity, acceleration, looping number, curvature and trajectory length, are non-objective, i.e., depend on the observer. To bridge this gap between LCS and sparse data, we derive measures of local material stretching and rotation that are computable from individual trajectories without reliance on other trajectories or on an underlying velocity field. Both measures are quasi-objective: they approximate objective (i.e., observer-independent) coherence diagnostics in frames satisfying a certain condition. We illustrate with several examples how our quasi-objective coherence diagnostics highlight elliptic and hyperbolic LCS, even from very sparse unstructured trajectory data. This approach shows great potential for expanding the possibilities of LCS applications through its simplicity, performance with sparse data, and enhanced computational efficiency.

Several procedures have been proposed to modify non-objective (observer-dependent) local vortex criteria so that they become objective. These modifications are only justifiable if they are equivalent to applying the original criteria after a generalized (possibly nonlinear) frame change is performed on the flow domain; otherwise, the arguments used in deriving those criteria no longer apply. To examine the feasibility of available objectivization procedures, we derive here necessary and sufficient conditions for the existence of a generalized frame change prescribed pointwise through its Jacobian field. From these conditions we conclude that, of all proposed objectivization approaches in the literature, only the replacement of the spin tensor with the spin-deviation tensor is applicable to generic fluid flows.

We derive universal upper estimates for model prediction error under moderate but otherwise unknown model uncertainty. Our estimates give upper bounds on the leading-order trajectory uncertainty arising along model trajectories, solely as functions of the invariants of the known Cauchy-Green strain tensor of the model. Our bounds turn out to be optimal, which means that they cannot be improved for general systems. The quantity relating the leading-order trajectory-uncertainty to the model uncertainty is the model sensitivity (MS), which we find to be a useful tool for a quick global assessment of the impact of modeling uncertainties in various domains of the phase space. By examining the expectation that finite-time Lyapunov exponents capture sensitivity to modeling errors, we show that this does not generally follow. However, we find that certain important features of the finite-time Lyapunov exponent persist in the MS field.

We derive a theory for material surfaces that maximally inhibit the diffusive transport of a dynamically active vector field, such as the linear momentum, the angular momentum or the vorticity, in general fluid flows. These special material surfaces (\emph{Lagrangian active barriers}) provide physics-based, observer-independent boundaries of dynamically active coherent structures. We find that Lagrangian active barriers evolve from invariant surfaces of an associated steady and incompressible \emph{barrier equation}, whose right-hand side is the time-averaged pullback of the viscous stress terms in the evolution equation for the dynamically active vector field. Instantaneous limits of these barriers mark objective \emph{Eulerian active barriers} to the short-term diffusive transport of the dynamically active vector field. We obtain that in unsteady Beltrami flows, Lagrangian and Eulerian active barriers coincide exactly with purely advective transport barriers bounding observed coherent structures. In more general flows, active barriers can be identified by applying Lagrangian coherent structure (LCS) diagnostics, such as the finite-time Lyapunov exponent and the polar rotation angle, to the appropriate active barrier equation. In comparison to their passive counterparts, these \emph{active LCS diagnostics} require no significant fluid particle separation and hence provide substantially higher-resolved Lagrangian and Eulerian coherent structure boundaries from temporally shorter velocity data sets. We illustrate these results and their physical interpretation on two-dimensional, homogeneous, isotropic turbulence and on a three-dimensional turbulent channel flow., 50 pages, 40 figures, to be published in J. Fluid. Mech

For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced model, no coordinate changes are required beyond an initial linear one. The reduced-order models we derive are simple and depend only on physical and modal parameters, allowing us to extract fundamental characteristics, such as backbone curves and forced-response curves, of multi-degree-of-freedom mechanical systems. To numerically verify the accuracy of the reduced models, we test the reduction formulas on several mechanical systems, including a higher-dimensional nonlinear Timoshenko beam.

In this study we evaluate weekly EASE-GRID sea-ice velocities released by the National Snow and Ice Data Centre (NSIDC). We identify persistent Eulerian and Lagrangian features, that arise solely as an artifact of the method used in the incorporation of buoy data. This, in turn, significantly impacts calculation of sea ice motion gradients, including divergence, convergence and shear. Our numerical experiments and comparison with observations further demonstrate the impact of these artificial features on climatological assessments, including age of ice studies. In particular, we find that age of ice studies using this dataset significantly underestimate multi-year ice extent by an average of 0.8 million km2 in the month of March.

Significance The detection of vortices in geophysical flows is important because of their role in global circulation and transport. The recent theory of elliptic Lagrangian coherent structures has shown that long-lived, coherent material eddies exhibiting no filamentation can be extracted from altimetry-derived surface velocities associated with slow, mesoscale motions in the ocean. Since similarly perfect eddies may not be expected to exist in the presence of faster, submesoscale motions, it has remained unclear whether persistent and materially coherent eddies can in fact be identified objectively even in highly resolved ocean data. Here we show that under a suitable extension of the notion of material coherence, coherent eddies at the mesoscale and submesoscale level can indeed be simultaneously detected from multiscale velocity data.

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.

[1] Application of recent geometric tools for Lagrangian coherent structures (LCS) shows that material attraction in geostrophic velocities derived from altimetry data imposed an important constraint to the motion of drifters from the Grand Lagrangian Deployment (GLAD) in the Gulf of Mexico. This material attraction is largely transparent to traditional Eulerian analysis. Attracting LCS acted as approximate centerpieces for mesoscale patterns formed by the drifters. Persistently attracting LCS cores emerged 1 week before the development of a filament resembling the “tiger tail” of the Deepwater Horizon oil slick, thereby anticipating its formation. Our results suggest that the mesoscale circulation plays a significant role in shaping near-surface transport in the Gulf of Mexico.

Recent studies have shown that aerial disturbances affecting landing aircraft have a coherent signature in the Lagrangian aerial particle dynamics inferred from ground-based lidar scans. Specifically, attracting Lagrangian coherent structures (LCSs) mark the intersection of localized material upwelling within the cone of the lidar scan. This study tests the detection power of LCSs on historical landing data and corresponding pilot reports of disturbances from Hong Kong International Airport. The results show that a specific LCS indicator, the gradient of the finite-time Lyapunov exponent (FTLE) field along the landing path, is a highly efficient marker of turbulent upwellings. In particular, in the spring season, projected FTLE gradients closely approach the efficiency of the wind shear alert system currently in operation at the airport, even though the latter system relies on multiple sources of data beyond those used in this study. This shows significant potential for the operational use of FTLE gradients in the real-time detection of aerial disturbances over airports.

Using observational data from coherent Doppler light detection and ranging (lidar) systems situated at the Hong Kong International Airport (HKIA), the authors extract Lagrangian coherent structures (LCS) intersecting the flight path of landing aircraft. They study the time evolution of LCS and compare them with onboard wind shear and altitude data collected during airplane approaches. Their results show good correlation between LCS extracted from the lidar data and updrafts and downdrafts experienced by landing aircraft. Overall, LCS analysis shows promise as a robust real-time tool to detect unsteady flow structures that impact airplane traffic.

Direct Lyapunov exponents and stability results are used to extract and distinguish Lagrangian coherent structures (LCS) from a three-dimensional atmospheric dataset generated from the Weather Research and Forecasting (WRF) model. The numerical model is centered at 19.78°N, 155.55°W, initialized from the Global Forecast System for the case of a subtropical jet stream near Hawaii on 12 December 2002. The LCS are identified that appear to create optical and mechanical turbulence, as evidenced by balloon data collected during a measurement campaign near Hawaii.

The phase space structure in the vicinity of a rank-two saddle is explored both analytically and numerically. In particular, the geometry of electron dynamics in the neighborhood of the rank-two saddle associated with the nonsequentual double ionization of helium is analyzed. As in the rank-one saddle case, codimension-one normally hyperbolic invariant manifolds turn out to control the rates of the correlated electron dynamics within each energy surface. The construction of these manifolds, however, is more involved here and requires the use of pseudo-hyperbolic invariant manifold theory. Two distinct correlated motions occur, the nonsequential double ionization and the nonsequential exchange of electrons. The relative rates of crossing the barrier are related to the Lyapunov exponents. The dynamics associated with the larger of the two Lyapunov exponents predominates.

We derive a simple criterion for transverse instabilities along a general invariant manifold of a multi- dimensional dynamical system. The criterion requires an appropriately defined normal infinitesimal Lyapunov exponent (NILE) to be positive over regions of transverse instability on the manifold. Unlike classic Lyapunov-type numbers in the theory of normally hyperbolic invariant manifolds, the NILE can be computed analytically in applications. This enables us to locate, for example, regions of transient jumping along an invariant manifold that is otherwise globally attracting. To illustrate our results, we determine the locations of intermittent instabilities in bubble motion past a cylinder, in predator-prey interactions, and in soft-stiff structural systems.

The motion of inertial (i.e., finite-size) particles is analyzed in a three-dimensional unsteady simulation of Hurricane Isabel. As established recently, the long-term dynamics of inertial particles in a fluid is governed by a reduced-order inertial equation, obtained as a small perturbation of passive fluid advection on a globally attracting slow manifold in the phase space of particle motions. Use of the inertial equation enables the visualization of three-dimensional inertial Lagrangian coherent structures (ILCS) on the slow manifold. These ILCS govern the asymptotic behavior of finite-size particles within a hurricane. A comparison of the attracting ILCS with conventional Eulerian fields reveals the Lagrangian footprint of the hurricane eyewall and of a large rainband. By contrast, repelling ILCS within the eye region admit a more complex geometry that cannot be compared directly with Eulerian features.

We apply a recent analytic theory of three-dimensional steady separation to direct numerical simulations of backward-facing-step and lid-driven-cavity flows. We determine the exact location and slope of separation and attachment surfaces that have only been approximated heuristically in earlier studies. We also visualize the corresponding global separation and attachment surfaces, which reveal highly complex separated flow geometry.

We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

We define objective Eulerian Coherent Structures (OECSs) in two-dimensional, non-autonomous dynamical systems as the instantaneously most influential material curves. Specifically, OECSs are stationary curves of the averaged instantaneous material stretching-rate or material shearing-rate functionals. From these objective (frame-invariant) variational principles, we obtain explicit differential equations for hyperbolic, elliptic, and parabolic OECSs. As an illustration, we compute OECSs in an unsteady ocean velocity data set. In comparison to structures suggested by other common Eulerian diagnostic tools, we find OECSs to be the correct short-term cores of observed trajectory deformation patterns.

In this paper, a study of the global dynamics of an autoparametric four degree-of-freedom (DOF) spring–mass–pendulum system with a rigid body mode is presented. Following a modal decoupling procedure, typical approximate periodic solutions are obtained for the autoparametrically coupled modes in 1:2 internal resonance. A novel technique based on forward-time solutions for finite-time Lyapunov exponent is used to establish global convergence and domains of attraction of different solutions. The results are compared to numerically constructed domains of attraction in the plane of initial position and initial velocity for the pendulum. Simulations are also provided for a few interesting cases of interest near critical values of parameters. Results also shed some light on the role played by other modes present in a multi-DOF system in shaping the overall system response.

Recent developments in identifying Lagrangian coherent structures from finite-time velocity data have provided a theoretical basis for understanding chaotic transport in general flows with aperiodic dependence on time. As these theoretical developments are extended and applied to more complex flows, an accurate and general numerical method for computing these structures is needed to exploit these ideas for engineering applications. We present an unstructured high-order hp/spectral-element method for solving the two-dimensional compressible form of the Navier–Stokes equations. A corresponding high-order particle tracking method is also developed for extracting the Lagrangian coherent structures from the numerically computed velocity fields. Two different techniques are used; the first computes the direct Lyapunov exponent from an unstructured initial particle distribution, providing easier resolution of structures located close to physical boundaries, whereas the second advects a small material line initialized close to a Lagrangian saddle point to delineate these structures. We demonstrate our algorithm on simulations of a bluff-body flow at a Reynolds number of Re = 150 and a Mach number of M = 0.2 with and without flow forcing. We show that, in the unforced flow, periodic vortex shedding is predicted by our numerical simulations that is in stark contrast to the aperiodic flow field in the case with forcing. An analysis of the Lagrangian structures reveals a transport barrier that inhibits cross-wake transport in the unforced flow. The transport barrier is broken with forcing, producing enhanced transport properties by chaotic advection and consequently improved mixing of advected scalars within the wake.

High-resolution ocean velocity data has become readily available since the introduction of very high frequency (VHF) radar technology. The vast amount of data generated so far, however, remains largely unused in environmental prediction. In this paper, we use VHF data of the Florida coastline to locate Lagrangian coherent structures (LCS) hidden in ocean surface currents. Such structures govern the spread of organic contaminants and passive drifters that stay confined to the ocean surface. We use the Lagrangian structures in a real-time pollution release scheme that reduces the effect of industrial contamination on the coastal environment.

Lagrangian Coherent Structures (LCSs) are special material surfaces that delineate tracer patterns in unsteady flows. The recently developed variational theory of LCSs enables their objective (frame-invariant) detection in numerical and experimental velocity data. Here we review the main results of this variational theory for elliptic LCSs (i.e., perfectly coherent material vortices) and show how such structures can be extracted from geophysical data sets.

Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.

This paper examines whether hyperbolic Lagrangian structures—such as stable and unstable manifolds—found in model velocity data represent reliable predictions for mixing in the true fluid velocity field. The error between the model and the true velocity field may result from velocity interpolation, extrapolation, measurement imprecisions, or any other deterministic source. We find that even large velocity errors lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. More specifically, we show how model predictions from the Okubo–Weiss criterion or from finite-time Lyapunov exponents can be validated. We also estimate how close the true Lagrangian coherent structures are to those predicted by models.

We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.

We introduce a Lagrangian definition for the boundaries of coherent structures in two-dimensional turbulence. The boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors. Such material lines are responsible for stretching and folding in the mixing of passive tracers. We derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets. The criterion provides a rigorous link between the Lagrangian concept of hyperbolicity, the Okubo–Weiss criterion, and vortex boundaries. We apply the results to simulations of two-dimensional barotropic turbulence.

The relationship between statistical and geometric properties of particle motion in aperiodic, two-dimensional flows is examined. Finite-time-invariant manifolds associated with transient hyperbolic trajectories are shown to divide the flow into distinct regions with similar statistical behavior. In particular, numerical simulations of simple, eddy-resolving barotropic flows indicate that there exists a close correlation between such geometric structures and patchiness plots that describe the distribution of Lagrangian average velocity over initial conditions. For barotropic turbulence, we find that Eulerian velocity correlation time scales are significantly longer than their Lagrangian counterparts indicating the existence of well-defined Lagrangian structures. Identification of such structures shows a similar, close relationship between the invariant manifold geometry and patchiness calculations at intermediate time scales, where anomalous dispersion rates are found.

We study the transport of particles in a general, two-dimensional, incompressible flow in the presence of a transient eddy, i.e., a bounded set of closed streamlines with a finite time of existence. Using quantities obtained from Eulerian observations, we provide explicit conditions for the existence of a hyperbolic structure in the flow, which induces mixing between the eddy and its environment. Our results can be used directly to study finite-time transport in numerically or experimentally generated vector fields with general time-dependence.

We develop a general, coordinate-free theory for the reduction of volume-preserving flows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.

Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step., 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left) that appears in the version published in Chaos

Nonlinear Dynamics, 73 (1-2), ISSN:0924-090X, ISSN:1573-269X

We propose here the use of the variational level set methodology to capture Lagrangian vortex boundaries in 2D unsteady velocity fields. This method reformulates earlier approaches that seek material vortex boundaries as extremum solutions of variational problems. We demonstrate the performance of this technique for two different variational formulations built upon different notions of coherence. The first formulation uses an energy functional that penalizes the deviation of a closed material line from piecewise uniform stretching [Haller and Beron-Vera, J. Fluid Mech. 731, R4 (2013)]. The second energy function is derived for a graph-based approach to vortex boundary detection [Hadjighasem et al., Phys. Rev. E 93, 063107 (2016)]. Our level-set formulation captures an a priori unknown number of vortices simultaneously at relatively low computational cost. We illustrate the approach by identifying vortices from different coherence principles in several examples.

Lagrangian coherent structures (LCSs) are material surfaces that shape finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from different variational principles, solving different equations. Here we observe that, in three dimensions, initial positions of all variational LCSs are invariant manifolds of the same autonomous dynamical system, generated by the intermediate eigenvector field, $\xi_{2}(x_{0})$, of the Cauchy-Green strain tensor. This $\xi_{2}$-system allows for the detection of LCSs in any unsteady flow by classic methods, such as Poincar\'e maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual $\xi_{2}$-system to uncover both hyperbolic and elliptic LCSs from a single computation.

In this paper we prove the existence of multi-pulse orbits homoclinic to a slow manifold in a two-mode truncation of the damped-forced nonlinear Schrodinger equation (first suggested by Bishop et al.). These orbits are jumping , i.e., the corresponding solutions keep switching in time between neighborhoods of the two charateristic “breathers” of the integrable limit. In the case of no damping, we find multi-pulse Smale horseshoes in the two-mode model, while in the dissipative case we establish the existence of structurally stable, multi-pulse, heterolinic connections between two unstable equilibria. The orbits we construct are not amenable to Melnikov-type perturbation methods. In both the Hamiltonian and the dissipative case we find homoclinic trees , which describe the repeated bifurcations of multi-pulse solutions. To illustrate the theoretical predictions, we also present visualizations of these complicated structures.

In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.

In this talk I will illustrate the geometry in phase space in the vicinity of a rank‐two saddle using an example of current experimental interest, the non‐sequential double ionization of helium. In particular, I will focus on the invariant manifolds which direct the Hamiltonian flow and on the transition states that enable one to calculate the rate of the flow. A full discussion of this work can be found in Ref. [1, 2].

The article of record as published may be found at http://dx.doi.org/10.1021/es0630691 High-frequency (HF) radar technology produces detailed velocity maps near the surface of estuaries and bays. The use of velocity data in environmental prediction, nonetheless, remains unexplored. In this paper, we uncover a striking flow structure in coastal radar observations of Monterey Bay, along the California coastline. This complex structure governs the spread of organic contaminants, such as agricultural runoff which is a typical source of pollution in the bay. We show that a HF radar-based pollution release scheme using this flow structure reduces the impact of pollution on the coastal environment in the bay. We predict the motion of the Lagrangian flow structures from finite-time Lyapunov exponents of the coastal HF velocity data. From this prediction, we obtain optimal release times, at which pollution leaves the bay most efficiently. Office of Naval Research grant N00014-01-1-0208 ASAP MURI N00014-02-1-0826 Office of Naval Research grant N00014-01-1-0208 ASAP MURI N00014-02-1-0826

Undulatory swimmers flex their bodies to displace water, and in turn, the flow feeds back into the dynamics of the swimmer. At moderate Reynolds number, the resulting flow structures are characterized by unsteady separation and alternating vortices in the wake. We use the flow field from simulations of a two-dimensional, incompressible viscous flow of an undulatory, self-propelled swimmer and detect the coherent Lagrangian vortices in the wake to dissect the driving momentum transfer mechanisms. The detected material vortex boundary encloses a Lagrangian control volume that serves to track back the vortex fluid and record its circulation and momentum history. We consider two swimming modes: the C-start escape and steady anguilliform swimming. The backward advection of the coherent Lagrangian vortices elucidates the geometry of the vorticity field and allows for monitoring the gain and decay of circulation and momentum transfer in the flow field. For steady swimming, momentum oscillations of the fish can largely be attributed to the momentum exchange with the vortex fluid. For the C-start, an additionally defined jet fluid region turns out to balance the high momentum change of the fish during the rapid start.

We present a technique that uncovers the Lagrangian building blocks of turbulence, and apply this technique to a quasi-two-dimensional turbulent flow experiment. Our analysis identifies an intricate network of attracting and repelling material lines. This chaotic tangle, the Lagrangian skeleton of turbulence, shows a level of complexity found previously only in theoretical and numerical examples of strange attractors. We quantify the strength (hyperbolicity) of each material line in the skeleton and demonstrate dramatically different mixing properties in different parts of the tangle.

We use a dynamical systems approach to extend Prandtl's steady separation criterion to two-dimensional unsteady flows with no-slip boundaries. Viewing separation profiles as non-hyperbolic unstable manifolds in the Lagrangian frame, we obtain explicit Eulerian formulae for the location of flow separation and reattachment on fixed and moving boundaries. We also derive high-order approximations for the unsteady separation profile in the vicinity of the boundary. Our criteria and formulae only use the derivatives of the velocity field along the boundary, and hence are of use in monitoring and controlling separation. In particular, we predict unsteady flow separation points and separation angles from distributed pressure and skin-friction measurements along the wall. As an example, we predict and verify separation points and separation profiles in variants of a two-dimensional oscillating separation-bubble flow.

where τ is the skin friction, ρ is the density, ν is the kinematic viscosity, and u(x,y) is the horizontal component of the velocity field. In geometric terms, (p,0) is the location where a singular streamline separates from the boundary. By contrast, fluid particles in unsteady flow separate from the boundary at points of nonzero skin friction. As an example, Fig. 1 shows the breakaway of fluid particles from the boundary at a point different from the instantaneous zero skin friction point, which is at 1.6 x = − . In general, as Sears and Tellionis observe, zero skin friction “does not denote separation in any meaningful sense in unsteady flow.” Instead, they propose, the Moore–Rott–Sears (MRS) principle should apply, by which separation takes place at offboundary points where the horizontal shear is zero, and the fluid velocity is equal to the velocity of the separation point. This postulate, however, assumes the a priori knowledge of the separation point, and hence appears inapplicable in practice. 5,6 Van Dommelen and co-workers notice that despite the lack of a well-defined separation location in the unsteady Eulerian frame, fluid particles do break away in a thin spike from the boundary in

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.

For two-dimensional velocity fields defined on finite time intervals, we derive an analytic condition that can be used to determine numerically the location of uniformly hyperbolic trajectories. The conditions of our main theorem will be satisfied for typical velocity fields in fluid dynamics where the deformation rate of coherent structures is slower than individual particle speeds. We also propose and test a simple numerical algorithm that isolates uniformly finite-time hyperbolic sets in such velocity fields. Uniformly hyperbolic sets serve as the key building blocks of Lagrangian mixing geometry in applications. (c) 2000 American Institute of Physics.

Many recent studies have been performed on resonantly excited mass-pendulum systems with autoparametric (internal) resonance capturing interesting local steady state phenomena. The objective of this work is to explore the transient behavior in such systems. The domains of attraction of the time-periodic system provide some help in understanding the transient dynamics, and these are sought using a recently developed algorithm that solves for the finite-time Lyapunov exponent over a grid of initial conditions. Though the use of finite-time Lyapunov exponents in nonlinear dynamical analyses is not novel, its application to multi-degree-offreedom forced nonlinear systems has not been reported in the literature. In addition to identifying regions of different final states, the technique used captures different levels of attraction within a domain. This sheds some light on the role played by other modes present in a multi-degree-of-freedom system in shaping the overall system response.Copyright © 2003 by ASME

Coherent boundaries of Lagrangian vortices in fluid flows have recently been identified as closed orbits of line fields associated with the Cauchy-Green strain tensor. Here we develop a fully automated procedure for the detection of such closed orbits in large-scale velocity data sets. We illustrate the power of our method on ocean surface velocities derived from satellite altimetry., 17 pages, 6 figures, 1 table, accepted for publication in Proc. R. Soc. Lond., Ser. A

Normally hyperbolic invariant manifolds (NHIMs) are well-known organizing centers of the dynamics in the phase space of a nonlinear system. Locating such manifolds in systems far from symmetric or integrable, however, has been an outstanding challenge. Here, we develop an automated detection method for codimension-one NHIMs in autonomous dynamical systems. Our method utilizes Stationary Lagrangian Coherent Structures (SLCSs), which are hypersurfaces satisfying one of the necessary conditions of a hyperbolic LCS, and are also quasi-invariant in a well-defined sense. Computing SLCSs provides a quick way to uncover NHIMs with high accuracy. As an illustration, we use SLCSs to locate two-dimensional stable and unstable manifolds of hyperbolic periodic orbits in the classic ABC flow, a three-dimensional solution of the steady Euler equations.

We study the validity of various models for the dynamics of finite-sized particles in fluids by means of a direct comparison between theory and experimentally measured trajectories and velocities of large numbers of particles in chaotic two-dimensional flow. Our analysis indicates that finite-sized particles follow the predicted particle dynamics given by the Maxey-Riley equation, except for random correlated fluctuations that are not captured by deterministic terms in the equations of motion, such as the Basset-Boussinesq term or the lift force. We describe the fluctuations via spectral methods and we propose three different Lagrangian stochastic models to account for them. These Lagrangian models are stochastic generalizations of the Maxey-Riley equation with coefficients calibrated to the experimental data. We find that one of them is capable of describing the observed fluctuations fairly well, while it also predicts a drag coefficient in near agreement with the theoretical Stokes drag.