Daniel Blazevski

Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows

Abstract We develop a general theory of transport barriers for three-dimensional unsteady flows with arbitrary time-dependence. The barriers are obtained as two-dimensional Lagrangian Coherent Structures (LCSs) that create locally maximal deformation. Along hyperbolic LCSs, this deformation is induced by locally maximal normal repulsion or attraction. Along shear LCSs, the deformation is created by locally maximal tangential shear. Hyperbolic LCSs, therefore, play the role of generalized stable and unstable manifolds, while closed shear LCSs (elliptic LCSs) act as generalized KAM tori or KAM-type cylinders. All these barriers can be computed from our theory as explicitly parametrized surfaces. We illustrate our results by visualizing two-dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the ABC flow.

Shearless transport barriers in unsteady two-dimensional flows and maps

Abstract We develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two-dimensional flows and maps defined over a finite time interval. This principle reveals that hyperbolic Lagrangian Coherent Structures (LCSs) and parabolic LCSs (or jet cores) are the two main types of shearless barriers in unsteady flows. Based on the boundary conditions they satisfy, parabolic barriers are found to be more observable and robust than hyperbolic barriers, confirming widespread numerical observations. Both types of barriers are special null-geodesics of an appropriate Lorentzian metric derived from the Cauchy–Green strain tensor. Using this fact, we devise an algorithm for the automated computation of parabolic barriers. We illustrate our detection method on steady and unsteady non-twist maps and on the aperiodically forced Bickley jet.

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 Jupiters 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.

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.

Heat pulse propagation in chaotic three-dimensional magnetic fields

Heat pulse propagation in three-dimensional chaotic magnetic fields is studied by solving numerically the parallel heat transport equation using a Lagrangian Greens function (LG) method. The LG method provides an efficient and accurate technique that circumvents known limitations of finite elements and finite difference methods. The main two problems addressed are (i) the dependence of the radial transport of heat pulses on the level of magnetic field stochasticity (controlled by the amplitude of the magnetic field perturbation, ), and (ii) the role of reversed shear magnetic field configurations on heat pulse propagation. In all the cases considered there are no magnetic flux surfaces. However, the radial transport of heat pulses is observed to depend strongly on due to the presence of high-order magnetic islands and Cantori. These structures act as quasi-transport barriers which can actually preclude the radial penetration of heat pulses within physically relevant time scales. The dependence of the magnetic field connection length, lB, on is studied in detail. Regions where lB is large, correlate with regions where the radial propagation of the heat pulse slows down or stops. The decay rate of the temperature maximum, 〈T〉max(t), the time delay of the temperature response as function of the radius, τ, and the radial heat flux , are also studied as functions of the magnetic field stochasticity and lB. In all cases it is observed that the scaling of 〈T〉max with t transitions from sub-diffusive, 〈T〉max ~ t−1/4, at short times (χ∥t 105). A strong dependence on is also observed on τ and . Even in the case when there are no flux surfaces nor magnetic field islands, reversed shear magnetic field configurations exhibit unique transport properties. The radial propagation of heat pulses in fully chaotic fields considerably slows down in the shear reversal region and, as a result, the delay time of the temperature response in reversed shear configurations is about an order of magnitude longer than the one observed in monotonic q-profiles. The role of separatrix reconnection of resonant modes in the shear reversal region, and the role of shearless Cantori in the observed phenomena are also discussed.