Effcts of smooth divergence-free flows on tracer gradients and spectra: Eulerian prognosis description

Ocean eddies play an important role in the transport of heat, salt, nutrients or pollutants. During a finite-time advection, the gradients of these tracers can increase or decrease, depending on a growth rate and the angle between flow gradients and initial tracer gradients. The growth rate is directly related to finite-time Lyapunov exponents. Numerous studies on mixing and/or tracer downscaling methods rely on satellite altimeter-derived ocean velocities. Filtering most oceanic small-scale eddies, the resulting smooth Eulerian velocities are often stationary during the characteristic time of tracer gradient growth. While smooth, these velocity fields are still locally misaligned, and thus uncorrelated, to many coarse-scale tracers observations amendable to downscaling (e.g. SST, SSS). Using finite-time advections, the averaged squared norm of tracer gradients can then only increase, with local growth rate independent of the initial coarse-scale tracer distribution. The key mixing processes are then only governed by locally uniform shears and foldings around stationary convective cells. To predict the tracer deformations and the evolution of their 2nd-order statistics, an effcient proxy is proposed. Applied to a single velocity snapshot, this proxy extends the Okubo-Weiss criterion. For the Lagrangian-advection-based downscaling methods, it further successfully predicts the evolution of tracer spectral energy density after a finite time, and the optimal time to stop the downscaling operation. A practical estimation can then be proposed to define an effective parameterization of the horizontal eddy diffusivity.

Do coherent structures organize scalar mixing in a turbulent boundary layer?

A scalar emanating from a point source in a turbulent boundary layer does not mix homogeneously, but is organized in large regions with little variation of the concentration: uniform concentration zones. We measure scalar concentration using laser-induced fluorescence and, simultaneously, the three-dimensional velocity field using tomographic particle image velocimetry in a water tunnel boundary layer. We identify uniform concentration zones using both a simple histogram technique, and more advanced cluster analysis. From the complete information on the turbulent velocity field, we compute two candidate velocity structures that may form the boundaries between two uniform concentration zones. One of these structures is related to the rate of point separation along Lagrangian trajectories and the other one involves the magnitude of strong shear in snapshots of the velocity field. Therefore, the first method allows for the history of the flow field to be monitored, while the second method only looks at a snapshot. The separation of fluid parcels in time was measured in two ways: the exponential growth of the separation as time progresses (related to finite-time Lyapunov exponents and unstable manifolds in the theory of dynamical systems), and the exponential growth as time moves backward (stable manifolds). Of these two, a correlation with the edges of uniform concentration zones was found for the past Lyapunov field but not with the time-forward future field. The magnitude of the correlation is comparable to that of the regions of strong shear in the instantaneous velocity field.

Lagrangian approach for modal analysis of fluid flows

Common modal decomposition techniques for flow-field analysis, data-driven modelling and flow control, such as proper orthogonal decomposition and dynamic mode decomposition, are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid–structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of proper orthogonal decomposition and dynamic mode decomposition using direct numerical simulations of two canonical flow configurations at Mach 0.5, i.e. the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that application of LMA to steady non-uniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents. We examine the mathematical link between finite-time Lyapunov exponents and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.

Hidden Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

Lagrangian betweenness: detecting fluid transport bottlenecks in oceanic flows

<p> </p><p>The study of connectivity patterns in networks has brought novel insights across diverse fields ranging from neurosciences to epidemic spreading or climate. In this context, betweenness centrality has demonstrated to be a very effective measure to identify nodes that act as focus of congestion, or bottlenecks, in the network. However, there is not a way to define betweenness outside the network framework. Here we introduce the “Lagrangian betweenness”, an analogous quantity which relies only on the information provided by trajectories sampled across a generic dynamical system in the form of Finite Time Lyapunov Exponents, a widely used metric in Dynamical Systems Theory and Lagrangian oceanography. Our theoretical framework reveals a link between regions of high betweenness and the hyperbolic behavior of trajectories in the system. For example, it identifies bottlenecks in fluid flows where particles are first brought together and then widely dispersed. This has many potential applications including marine ecology and pollutant dispersal. We first test our definition of betweenness in an idealized double-gyre flow system. We then apply it in the characterization of transport by real geophysical flows in the semi-enclosed Adriatic Sea and the Kerguelen region of the highly turbulent Antarctic Circumpolar Current. In both cases, patterns of Lagrangian betweenness identify hidden bottlenecks of tracer transport that are surprisingly persistent across different spatio-temporal scales. In the marine context, high Lagrangian betweenness regions represent the optimal compromise between the heterogeneity of water origins and destinations, suggesting that they may be associated with relevant diversity reservoirs and hot-spots in marine ecosystems. Our new metric could also provide a novel approach useful for the management of environmental resources, informing strategies for marine spatial planning, and for designing observational networks to control pollutants or early-warning signals of climatic risks. </p>

Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.

The Influence of an Eddy in the Success Rates and Distributions of Passively Advected or Actively Swimming Biological Organisms Crossing the Continental Slope

The Lagrangian characteristics of the surface flow field arising when an idealized, anticyclonic, mesoscale, isolated deep-ocean eddy collides with continental slope and shelf topography are explored. In addition to fluid parcel trajectories, we consider the trajectories of biological organisms that are able to navigate and swim, and for which shallow water is a destination. Of particular interest is the movement of organisms initially located in the offshore eddy, the manner in which the eddy influences the ability of the organisms to reach the shelf break, and the spatial and temporal distributions of organisms that do so. For nonswimmers or very slow swimmers, the organisms arrive at the shelf break in distinct pulses, with different pulses occurring at different locations along the shelf break. This phenomenon is closely related to the episodic formation of trailing vortices that are formed after the eddy collides with the continental slope, turns, and travels parallel to the coast. Analysis based on finite-time Lyapunov exponents reveals initial locations of all successful trajectories reaching the shoreline, and provides maps of the transport pathways showing that much of the cross-shelf-break transport occurs in the lee of the eddy as it moves parallel to the shore. The same analysis shows that the onshore transport is interrupted after a trailing vortex detaches. As the swimming speeds are increased, the organisms are influenced less by the eddy and tend to show up en mass and in a single pulse.