The detection of coherent structures is an important problem in fluid dynamics, particularly in geophysical applications. For instance, knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills. Existing methods detect Lagrangian coherent structures, which are barriers to transport, by examining the stretching field as given by finite-time Lyapunov exponents. These methods are very effective when the velocity field is well-determined, but in many applications only a small number of flow trajectories are known, for example when dealing with oceanic float data. We introduce a topological method for detecting invariant regions based on a small set of trajectories. In the method we regard the two-dimensional trajectory data as a braid in three dimensions, with time being the third coordinate. Invariant regions then correspond to trajectories that travel together and do not entangle other trajectories. We detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories, and searching for loops that show negligible growth.

There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. The transport of material by advection in fluid systems is a process of vital and ubiquitous importance, underlying scenarios as diverse as pollutant distribution and searchand-rescue operations in the ocean to blood flow in the human body. The rapidly advancing field of Lagrangian based methods for studying flow transport has demonstrated an effective ability to find robust and significant transport features that underly the organization of flow transport in complex, unsteady flow fields. In this review, we present an overview of four of the leading Lagrangian approaches, each with their own strengths and challenges. Details of each method are presented along with an example application to the same model system, the double-gyre flow. Furthermore, we highlight a number of exciting applications and future directions to bring Lagrangian based analysis closer to implementation in real-time, real-world decision making strategies.

Using filter-space techniques, we study the scale-to-scale transport of energy in a quasi-two-dimensional, weakly turbulent fluid flow averaged along the trajectories of fluid elements. We find that although the spatial mean of this Lagrangian-averaged flux is nearly unchanged from its Eulerian counterpart, the spatial structure of the scale-to-scale energy flux changes significantly. In particular, its features appear to correlate with the positions of Lagrangian coherent structures (LCS's). We show that the LCS's tend to lie at zeros of the scale-to-scale flux, and therefore that the LCS's separate regions that have qualitatively different dynamics. Since LCS's are also known to be impenetrable barriers to advection and mixing, we therefore find that the fluid on either side of an LCS is both kinematically and dynamically distinct. Our results extend the utility of LCS's by making clear the role they play in the flow dynamics in addition to the kinematics.

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Buoyancy-driven flow, which is flow driven by spatial variations in fluid density 1 , lies at the heart of a variety of physical processes, including mineral transport in rocks 2 , the melting of icebergs 3 and the migration of tectonic plates 4 . Here we show that buoyancy-driven flows can also generate propulsion. Specifically, we find that when a neutrally buoyant wedgeshaped object floats in a density-stratified fluid, the diffusiondriven flow at its sloping boundaries generated by molecular diffusion produces a macroscopic sideways thrust. Computer simulations reveal that thrust results from diffusion-driven flow creating a region of low pressure at the front, relative to the rear of an object. This discovery has implications for transport processes in regions of varying fluid density, such as marine snow aggregation at ocean pycnoclines 5 , and wherever there is a temperature difference between immersed objects and the surrounding fluid, such as particles in volcanic clouds 6 .A fluid system with spatially varying density, resulting from temperature and/or salinity variations, for example, is stably-stratified when increasing density is parallel to the direction of gravity. When an object is released in a quiescent, stably-stratified fluid, it is expected to settle or rise to the neutral buoyancy level at which the density of the object matches that of the surrounding fluid, and remain stationary thereafter. We carried out a control experiment in which a 19.05-mm-diameter sphere of density ρ = 1,115 kg m −3 was released in a tank of height H = 0.40 m, width W = 0.20 m and length L = 0.40 m, filled with salt-stratified water with density gradient dρ/dz = −511 ± 3 kg m −4 . As expected, the sphere settled to its neutral buoyancy height, where it remained stationary for 24 hours. A similar experiment was then carried out using a triangular wedge ( Fig. 1) of length l = 99.9 ± 0.1 mm, base h = 17.6 ± 0.10 mm (corresponding to slope angle α = 5.0 ± 0.1 • ) and width w = 25.1 ± 0.10 mm. In striking contrast to the stationary sphere, the wedge moved at a constant speed u = 10.2 ± 0.1 × 10 −3 m h −1 (2.83 ± 0.03 × 10 −6 m s −1 ) in the direction of its tip, without any obvious cause ( Fig. 1 and its top inset; also see Supplementary Movie S1).To investigate the cause of this spontaneous propulsion, we visualized the velocity fields in the horizontal and vertical midplanes of the moving wedge using particle image velocimetry (PIV; Fig. 2). Here, z is the vertical coordinate antiparallel to gravity and x and y are the coordinates in the horizontal plane, parallel and perpendicular to the long axis of the wedge, respectively. These experiments reveal that fluid is drawn in towards the wedge tip in the horizontal plane to supply up-slope flow in a thin boundary layer above the sloping surface. Furthermore, fluid immediately behind the wedge moves with the same speed as the wedge; this phenomenon, known as blocking, occurs for obstacles in stratified flow when buoyancy forces dominate inertia and viscous forces 1 . As ...

Determination of energy transport is crucial for understanding the energy budget and fluid circulation in density varying fluids such as the ocean and the atmosphere. However, it is rarely possible to determine the energy flux field J = pu, which requires simultaneous measurements of the pressure and velocity perturbation fields, p and u. We present a method for obtaining the instantaneous J (x, z, t) from density perturbations alone: a Green's function-based calculation yields p, and u is obtained by integrating the continuity equation and the incompressibility condition. We validate our method with results from Navier-Stokes simulations: the Green's function method is applied to the density perturbation field from the simulations, and the result for J is found to agree typically to within 1% with J computed directly using p and u from the Navier-Stokes simulation. We also apply the Green's function method to density perturbation data from laboratory schlieren measurements of internal waves in a stratified fluid, and the result for J agrees to within 6% with results from Navier-Stokes simulations. Our method for determining the instantaneous velocity, pressure, and energy flux fields applies to any system described by a linear approximation of the density perturbation field, e.g., to small amplitude lee waves and propagating vertical modes. The method can be applied using our Matlab graphical user interface EnergyFlux.

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While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.