Search

Nonlinear normal modes are of great importance to understand the dynamical behavior of nonlinear mechanical systems and serve as natural candidates for model order reduction. Therefore, their accurate computation as well as mathematical foundation has extensively been studied. While numerous results are available for unforced and periodically forced nonlinear mechanical systems, the case of random external forcing is commonly not considered in the literature. Here, we clarify the relevance of nonlinear normal modes in the case of small white noise excitation. We demonstrate our results on explicit mechanical systems.

Nonlinear system identification is a challenging problem in experimental modal analysis. It is currently tackled using a toolbox approach, where different techniques are employed depending on the structural system under investigation, the identification goals and the type of excitation used. In this contribution, we exploit analytic reduction to spectral submanifolds combined with machine learning techniques in order to obtain the nonlinear coefficients up to cubic order of a single-degree-of-freedom reduced order model. The system measurements aimed at model fitting can be performed using any type of excitation techniques, ranging from free-decay to sine-sweeps or random shaker testing. We illustrate the accuracy of our method using both simulated and real experimental data.

The surface area of turbulent/non-turbulent interfaces (TNTIs) is continuously produced and destroyed via stretching and curvature/propagation effects. Here, the mechanisms responsible for TNTI area growth and destruction are investigated in a turbulent flow with and without stable stratification through the time evolution equation of the TNTI area. We show that both terms have broad distributions and may locally contribute to either production or destruction. On average, however, the area growth is driven by stretching, which is approximately balanced by destruction by the curvature/propagation term. To investigate the contribution of different length scales to these processes, we apply spatial filtering to the data. In doing so, we find that the averages of the stretching and the curvature/propagation terms balance out across spatial scales of TNTI wrinkles and this scale-by-scale balance is consistent with an observed scale invariance of the nearby coherent vortices. Through a conditional analysis, we demonstrate that the TNTI area production (destruction) is localized at the front (lee) edge of the vortical structures in the interface proximity. Finally, we show that while basic mechanisms remain the same, increasing stratification reduces the rates at which TNTI surface area is produced as well as destroyed. We provide evidence that this reduction is largely connected to a change in the multiscale geometry of the interface, which tends to flatten in the wall-normal direction at all active length scales of the TNTI.

The above mentioned article was submitted with incorrect author affiliations for George Haller and Markus Holzner. This has since been rectified in the online PDF and HTML copies. (J. Fluid Mech. (2020) 898 (A3) DOI: 10.1017/jfm.2020.414)

Every year, hundreds of people die at sea because of vessel and airplane accidents. A key challenge in reducing the number of these fatalities is to make Search and Rescue (SAR) algorithms more efficient. Here, we address this challenge by uncovering hidden TRansient Attracting Profiles (TRAPs) in ocean-surface velocity data. Computable from a single velocity-field snapshot, TRAPs act as short-term attractors for all floating objects. In three different ocean field experiments, we show that TRAPs computed from measured as well as modeled velocities attract deployed drifters and manikins emulating people fallen in the water. TRAPs, which remain hidden to prior flow diagnostics, thus provide critical information for hazard responses, such as SAR and oil spill containment, and hence have the potential to save lives and limit environmental disasters., Every year, hundreds of people die at sea because of vessel accidents, and a key challenge in reducing these fatalities is to make Search and Rescue (SAR) planning more efficient. Here, the authors uncover hidden flow features that attract floating objects, providing specific information for optimal SAR planning.

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on the exact curvature evolution of near-wall material lines, our theory identifies both fixed and moving flow separation, is effective also over short time intervals, and admits a rigorous instantaneous limit. As a byproduct, we derive explicit formulae for the evolution of material line curvature and the curvature rate for general compressible flows. The material backbone that we identify acts first as the precursor and later as the centrepiece of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified. We illustrate our results on several analytical and experimental flows.

We discuss an automated computational methodology for computing two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest nonlinear continuations of modal subspaces of the linearized system, are constructed up to arbitrary orders of accuracy, using the parameterization method. An advantage of this approach is that the construction of the SSMs does not break down when the SSM folds over its underlying spectral subspace. A further advantage is an automated a posteriori error estimation feature that enables a systematic increase in the orders of the SSM computation until the required accuracy is reached. We find that the present algorithm provides a major speed-up, relative to numerical continuation methods, in the computation of backbone curves, especially in higher-dimensional problems. We illustrate the accuracy and speed of the automated SSM algorithm on lower- and higher-dimensional mechanical systems.

The polar vortices play a crucial role in the formation of the ozone hole and can cause severe weather anomalies. Their boundaries, known as the vortex `edges', are typically identified via methods that are either frame-dependent or return non-material structures, and hence are unsuitable for assessing material transport barriers. Using two-dimensional velocity data on isentropic surfaces in the northern hemisphere, we show that elliptic Lagrangian Coherent Structures (LCSs) identify the correct outermost material surface dividing the coherent vortex core from the surrounding incoherent surf zone. Despite the purely kinematic construction of LCSs, we find a remarkable contrast in temperature and ozone concentration across the identified vortex boundary. We also show that potential vorticity-based methods, despite their simplicity, misidentify the correct extent of the vortex edge. Finally, exploiting the shrinkage of the vortex at various isentropic levels, we observe a trend in the magnitude of vertical motion inside the vortex which is consistent with previous results.

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the softer degrees of freedom. This slow–fast decomposition (SFD) enslaves exponentially fast the stiffer degrees of freedom to the softer ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justification for two modal reduction methods used in structural dynamics: static condensation and modal derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on mechanical examples.

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

Frequency responses of multi-degree-of-freedom mechanical systems with weak forcing and damping can be studied as perturbations from their conservative limit. Specifically, recent results show how bifurcations near resonances can be predicted analytically from conservative families of periodic orbits (nonlinear normal modes). However, the stability of forced-damped motions is generally determined a posteriori via numerical simulations. In this paper, we present analytic results on the stability of periodic orbits that perturb from conservative nonlinear normal modes. In contrast with prior approaches to the same problem, our method can tackle strongly nonlinear oscillations, high-order resonances, and arbitrary types of non-conservative forces affecting the system, as we show with specific examples.

We apply the recent frame-invariant theory of separation spike formation to complex unsteady flows, including a turbulent separation bubble, an impinging jet, and flows around a freely moving cylinder and a freely rotating ellipse. We show how the theory captures the onset of material spike formation, without any assumption on the flow type (steady, periodic, unsteady) or separation type (on- or off-wall, fixed or moving boundaries). We uncover new phenomena, such as the transition from on-wall to off-wall separation, the merger of initially distinct spikes, and the presence of severe material spikes that remain hidden to previous approaches. Remarkably, even in steady flows around curved boundaries, we detect material spikes in the absence of flow reversal, the main ingredient to existing separation criteria. Together, our results unveil how an involved network of spikes arises, interacts and merges dynamically, leading to the final ejection of particles from the wall in highly transient flow separation processes.

In this paper, we employ the theory of Lagrangian coherent structures for three-dimensional vortex eduction and investigate the effect of large-scale vortical structures on the turbulent/non-turbulent interface (TNTI) and entrainment of a gravity current. The gravity current is realized experimentally and different levels of stratification are examined. For flow measurements, we use a multivolume three-dimensional particle tracking velocimetry technique. To identify vortical Lagrangian coherent structures (VLCSs), a fully automated three-dimensional extraction algorithm for multiple flow structures based on the so-called Lagrangian-averaged vorticity deviation method is implemented. The size, the orientation and the shape of the VLCSs are analysed and the results show that these characteristics depend only weakly on the strength of the stratification. Through conditional analysis, we provide evidence that VLCSs modulate the average TNTI height, consequently affecting the entrainment process. Furthermore, VLCSs influence the local entrainment velocity and organize the flow field on both the turbulent and non-turbulent sides of the gravity current boundary.

We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a numerically generated turbulent Navier--Stokes flow.

Weakly damped mechanical systems under small periodic forcing tend to exhibit periodic response in a close vicinity of certain periodic orbits of their conservative limit. Specifically, amplitude-frequency plots for the conservative limit have often been noted, both numerically and experimentally, to serve as backbone curves for the near resonance peaks of the forced response. In other cases, such a relationship between the unforced and forced response was not observed. Here, we provide a systematic mathematical analysis that predicts which members of conservative periodic orbit families will serve as backbone curves for the forced–damped response. We also obtain mathematical conditions under which approximate numerical and experimental approaches, such as energy balance and force appropriation, are justifiable. Finally, we derive analytic criteria for the birth of isolated response branches (isolas) whose identification is otherwise challenging from numerical continuation.

Backbone curves are often the best representation of the nonlinear behavior for the vibrations of mechanical systems. Several approaches for obtaining them are present in literature, either analytical, numerical or experimental ones. However, they often make assumptions that unavoidably limit the range of applicability, such as the dynamics of the underlying conservative system and the modeling of damping terms. Here, we describe a mathematical theory and the corresponding numerical methodology that is able to rigorously extract backbone curves from free-decay vibrations data and that can overcome some of the main limitations of existing methods. We illustrate our findings with synthetic and real experiment vibration measurements.

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.

We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation, thus the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield., To appear in Nonlinear Dynamics

Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of spectral submanifolds (SSMs), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulae for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite-element models.

We show how spectral submanifold (SSM) theory can be used to extract forced-response curves without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems. We use multivariate recurrence relations to construct the SSMs, achieving a major speed-up relative to earlier autonomous SSM algorithms. The increase in computational efficiency promises to close the current gap between studying lower-dimensional academic examples and analyzing larger systems obtained from finite-element modeling, as we illustrate on two different discretized damped-forced beam models. Using the exact reduction procedure via SSMs for obtaining forced response curves, we further demonstrate speed gains of several orders in magnitude relative to the available state-of-the-art continuation packages, while retaining accuracy.

Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located directly from the deterministic component of the velocity field without diffusive or stochastic simulations. Here we extend these results to compressible flows and to diffusive concentration fields affected by sources or sinks, as well as by spontaneous decay. We construct diffusive transport extremizers with and without constraining them on a specific initial concentration distribution. For two-dimensional flows, we obtain explicit differential equations and a diagnostic scalar field that identify the most observable extremizers with pointwise uniform transport density. We illustrate our results by uncovering diffusion barriers and enhancers in analytic, numerical, and observational velocity fields., 33 pages, 6 figures, submitted

While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.

We construct a class of spatially polynomial velocity fields that are exact solutions of the planar unsteady Navier-Stokes equation. These solutions can be used as simple benchmarks for testing numerical methods or verifying the feasibility of flow-feature identification principles. We use examples from the constructed solution family to illustrate deficiencies of streamlines-based feature detection and of the Okubo-Weiss criterion, which is the common two-dimensional version of the broadly used Q-, Delta-, Lambda-2- and Lambda-Ci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier-Stokes solutions with a symmetry., Comment: To be published in Physics of Fluids

Inverting an evolving diffusive scalar field to reconstruct the underlying velocity field is an underdetermined problem. Here we show, however, that for two-dimensional incompressible flows, this inverse problem can still be uniquely solved if high-resolution tracer measurements, as well as velocity measurements along a curve transverse to the instantaneous scalar contours, are available. Such measurements enable solving a system of partial differential equations for the velocity components by the method of characteristics. If the value of the scalar diffusivity is known, then knowledge of just one velocity component along a transverse initial curve is sufficient. These conclusions extend to the shallow-water equations and to flows with spatially dependent diffusivity. We illustrate our results on velocity reconstruction from tracer fields for planar Navier- Stokes flows and for a barotropic ocean circulation model. We also discuss the use of the proposed velocity reconstruction in oceanographic applications to extend localised velocity measurements to larger spatial domains with the help of remotely sensed scalar fields., Comment: 20 pages, 10 figures, In press at J. Fluid Mechanics

ISSN:1042-8275 ISSN:2377-617X

We combine numerical simulations and an analytic approach to show that the capture of finite, inertial particles during flow in branching junctions is due to invisible, anchor-shaped three-dimensional flow structures. These Reynolds-number-dependent anchors define trapping regions that confine particles to the junction. For a wide range of Stokes numbers, these structures occupy a large part of the flow domain. For flow in a V-shaped junction, at a critical Stokes number, we observe a topological transition due to the merger of two anchors into one. From a stability analysis, we identify the parameter region of particle sizes and densities where capture due to anchors occurs.

To understand and analyze the behavior of realistic nonlinear structures it is desirable to reduce the dimensionality of the system, as well as simplify the equation of motion. Reduction to Spectral submanifolds (SSMs) has recently been shown to provide such a dimension reduction, yielding exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time-dependent SSMs. We demonstrate our analytical formulae on numerical examples and compare them to results obtained from alternative methods.

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.

We propose rotation inferred from the polar decomposition of the flow gradient as a diagnostic for elliptic (or vortex-type) invariant regions in non-autonomous dynamical systems. We consider here two- and three-dimensional systems, in which polar rotation can be characterized by a single angle. For this polar rotation angle (PRA), we derive explicit formulas using the singular values and vectors of the flow gradient. We find that closed level sets of the PRA reveal elliptic islands in great detail, and singular level sets of the PRA uncover centers of such islands. Both features turn out to be objective (frame-invariant) for two-dimensional systems. We illustrate the diagnostic power of PRA for elliptic structures on several examples.

The Maxey–Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing term that blows up at the initial time. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving local existence and uniqueness of mild solutions. For certain initial velocities between the particle and the fluid, the results extend to strong solutions. We also prove continuous differentiability of the mild and strong solutions with respect to their initial conditions. This justifies the search for coherent structures in inertial flows using the Cauchy–Green strain tensor.

We seek transport barriers and transport enhancers as material surfaces across which the transport of diffusive tracers is minimal or maximal in a general, unsteady flow. We find that such surfaces are extremizers of a universal, nondimensional transport functional whose leading-order term in the diffusivity can be computed directly from the flow velocity. The most observable (uniform) transport extremizers are explicitly computable as null surfaces of an objective transport tensor. Even in the limit of vanishing diffusivity, these surfaces differ from all previously identified coherent structures for purely advective fluid transport. Our results extend directly to stochastic velocity fields and hence enable transport barrier and enhancer detection under uncertainties., Proceedings of the National Academy of Sciences of the United States of America, 115 (37), ISSN:0027-8424, ISSN:1091-6490

We show how spectral submanifold (SSM) theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is becoming computationally expensive.

Typical fluid particle trajectories are sensitive to changes in their initial conditions. This makes the assessment of flow models and observations from individual tracer samples unreliable. Behind complex and sensitive tracer patterns, however, there exists a robust skeleton of material surfaces, Lagrangian coherent structures (LCSs), shaping those patterns. Free from the uncertainties of single trajectories, LCSs frame, quantify, and even forecast key aspects of material transport. Several diagnostic quantities have been proposed to visualize LCSs. More recent mathematical approaches identify LCSs precisely through their impact on fluid deformation. This review focuses on the latter developments, illustrating their applications to geophysical fluid dynamics.

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.

In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy., Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2202), ISSN:1364-5021, ISSN:1471-2946, ISSN:0080-4630, ISSN:0950-1207

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.

Mesoscale oceanic eddies are routinely detected from instantaneous velocities derived from satellite altimetry data. While simple to implement, this approach often gives spurious results and hides true material transport. Here it is shown how geodesic transport theory, a recently developed technique from nonlinear dynamical systems, uncovers eddies objectively. Applying this theory to altimetry-derived velocities in the South Atlantic reveals, for the first time, Agulhas rings that preserve their material coherence for several months, while ring candidates yielded by other approaches tend to disperse or leak within weeks. These findings suggest that available velocity-based estimates for the Agulhas leakage, as well as for its impact on ocean circulation and climate, need revision.

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here, we extend these results to periodically or quasi-periodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal-form methods.

We apply two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Karman beam with geometric nonlinearities and viscoelastic damping. SFD identifies a global slow manifold in the full system which attracts solutions at rates faster than typical rates within the manifold. An SSM, the smoothest nonlinear continuation of a linear modal subspace, is then used to further reduce the beam equations within the slow manifold. This two-stage, mathematically exact procedure results in a drastic reduction of the finite-element beam model to a one-degree-of freedom nonlinear oscillator. We also introduce the technique of spectral quotient analysis, which gives the number of modes relevant for reduction as output rather than input to the reduction process.

Rotationally coherent Lagrangian vortices (RCLVs) are identified from satellite-derived surface geostrophic velocities in the eastern Pacific (180°–130°W) using the objective (frame invariant) finite-time Lagrangian coherent structure detection method of Haller et al. based on the Lagrangian-averaged vorticity deviation. RCLVs are identified for 30-, 90-, and 270-day intervals over the entire satellite dataset, beginning in 1993. In contrast to structures identified using Eulerian eddy-tracking methods, the RCLVs maintain material coherence over the specified time intervals, making them suitable for material transport estimates. Statistics of RCLVs are compared to statistics of eddies identified from sea surface height (SSH) by Chelton et al. RCLVs and SSH eddies are found to propagate westward at similar speeds at each latitude, consistent with the Rossby wave dispersion relation. However, RCLVs are uniformly smaller and shorter-lived than SSH eddies. A coherent eddy diffusivity is derived to quantify the contribution of RCLVs to meridional transport; it is found that RCLVs contribute less than 1% to net meridional dispersion and diffusion in this sector, implying that eddy transport of tracers is mostly due to incoherent motions, such as swirling and filamentation outside of the eddy cores, rather than coherent meridional translation of eddies themselves. These findings call into question prior estimates of coherent eddy transport based on Eulerian eddy identification methods.

New techniques promise better forecasting of where damaging contaminants in the ocean or atmosphere will end up.

We introduce a new approach to locating key material transport barriers in two-dimensional, non-autonomous dynamical systems, such as unsteady planar fluid flows. Seeking transport barriers as minimally stretching material lines, we obtain that such barriers must be shadowed by minimal geodesics under the Riemannian metric induced by the Cauchy–Green strain tensor. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, we locate hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) in temporally aperiodic flows defined over a finite time interval. Our approach also yields a metric (geodesic deviation) that determines the minimal computational time scale needed for a robust numerical identification of generalized Lagrangian Coherent Structures (LCSs). As we show, an extension of our transport barrier theory to non-Euclidean flow domains, such as a sphere, follows directly. We illustrate our main results by computing key transport barriers in a chaotic advection map, and in a geophysical model flow with chaotic time dependence.

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.

We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the Cauchy–Green strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension-one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Cauchy–Green strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.