A critical comparison of Lagrangian methods for coherent structure detection

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.

Global variational approach to elliptic transport barriers in three dimensions

We introduce an approach to identify elliptic transport barriers in three-dimensional, time-aperiodic flows. Obtained as Lagrangian Coherent Structures (LCSs), the barriers are tubular non-filamenting surfaces that form and bound coherent material vortices. This extends a previous theory of elliptic LCSs as uniformly stretching material surfaces from two-dimensional to three-dimensional flows. Specifically, we obtain explicit expressions for the normals of pointwise (near-) uniformly stretching material surfaces over a finite time interval. We use this approach to visualize elliptic LCSs in steady and time-aperiodic ABC-type flows.

Local and nonlocal anisotropic transport in reversed shear magnetic fields: shearless Cantori and nondiffusive transport

A study of anisotropic heat transport in reversed shear (nonmonotonic q-profile) magnetic fields is presented. The approach is based on a recently proposed Lagrangian-Green's function method that allows an efficient and accurate integration of the parallel (i.e., along the magnetic field) heat transport equation. The magnetic field lines are described by a nontwist Hamiltonian system, known to exhibit separatrix reconnection and robust shearless (dq/dr=0) transport barriers. The changes in the magnetic field topology due to separatrix reconnection lead to bifurcations in the equilibrium temperature distribution. For perturbations of moderate amplitudes, magnetic chaos is restricted to bands flanking the shearless region. As a result, the temperature flattens in the chaotic bands and develops a very sharp radial gradient at the shearless region. For perturbations with larger amplitude, shearless Cantori (i.e., critical magnetic surfaces located at the minimum of the q profile) give rise to anomalous temperature relaxation involving widely different time scales. The first stage consists of the relatively fast flattening of the radial temperature profile in the chaotic bands with negligible flux across the shearless region that, for practical purposes, on a short time scale acts as an effective transport barrier despite the lack of magnetic flux surfaces. In the long-time scale, heat starts to flow across the shearless region, albeit at a comparatively low rate. The transport of a narrow temperature pulse centered at the reversed shear region exhibits weak self-similar scaling with non-Gaussian scaling functions indicating that transport at this scale cannot be modeled as a diffusive process with a constant diffusivity. Evidence of nonlocal effective radial transport is provided by the existence of regions with nonzero heat flux and zero temperature gradient. Parametric flux-gradient plots exhibit multivalued loops that question the applicability of the Fourier-Fick's prescription even in the presence of a finite pinch velocity.

Using scattering theory to compute invariant manifolds and numerical results for the laser-driven Hénon-Heiles system

Scattering theory is a convenient way to describe systems that are subject to time-dependent perturbations which are localized in time. Using scattering theory, one can compute time-dependent invariant objects for the perturbed system knowing the invariant objects of the unperturbed system. In this paper, we use scattering theory to give numerical computations of invariant manifolds appearing in laser-driven reactions. In this setting, invariant manifolds separate regions of phase space that lead to different outcomes of the reaction and can be used to compute reaction rates.