Lagrangian coherent structures are time-evolving surfaces that highlight areas in flow fields where neighboring advected particles diverge or converge. The detection and understanding of such structures is an important part of many applications such as in oceanography where there is a need to predict the dispersion of oil and other materials in the ocean. One of the most widely used tools for revealing Lagrangian coherent structures has been to calculate the finite-time Lyapunov exponents, whose maximal values appear as ridgelines to reveal Lagrangian coherent structures. In this paper we explore an alternative formulation of Lyapunov exponents for computing Lagrangian coherent structures.

Area-preserving maps are found across a wide range of scientific and engineering problems. Their study is made challenging by the significant computational effort typically required for their inspection but more fundamentally by the fractal complexity of salient structures. The visual inspection of these maps reveals a remarkable topological picture consisting of fixed (or periodic) points embedded in so-called island chains, invariant manifolds, and regions of ergodic behavior. This paper is concerned with the effective visualization and precise topological analysis of area-preserving maps with two degrees of freedom from numerical or analytical data. Specifically, a method is presented for the automatic extraction and characterization of fixed points and the computation of their invariant manifolds, also known as separatrices, to yield a complete picture of the structures present within the scale and complexity bounds selected by the user. This general approach offers a significant improvement over the visual representations that are so far available for area-preserving maps. The technique is demonstrated on a numerical simulation of magnetic confinement in a fusion reactor.

In the development of magnetic confinement fusion which will potentially be a future source for low cost power, physicists must be able to analyze the magnetic field that confines the burning plasma. While the magnetic field can be described as a vector field, traditional techniques for analyzing the field's topology cannot be used because of its Hamiltonian nature. In this paper we describe a technique developed as a collaboration between physicists and computer scientists that determines the topology of a toroidal magnetic field using fieldlines with near minimal lengths. More specifically, we analyze the Poincaré map of the sampled fieldlines in a Poincaré section including identifying critical points and other topological features of interest to physicists. The technique has been deployed into an interactive parallel visualization tool which physicists are using to gain new insight into simulations of magnetically confined burning plasmas.

Effective visualization of vector fields relies on the ability to control the size and density of the underlying mapping to visual cues used to represent the field. In this paper we introduce the use of a reaction-diffusion model, already well known for its ability to form irregular spatio-temporal patters, to control the size, density, and placement of the vector field representation. We demonstrate that it is possible to encode vector field information (orientation and magnitude) into the parameters governing a reaction-diffusion model to form a spot pattern with the correct orientation, size, and density, creating an effective visualization. To encode direction we texture the spots using a light to dark fading texture. We also show that it is possible to use the reaction-diffusion model to visualize an additional scalar value, such as the uncertainty in the orientation of the vector field. An additional benefit of the reaction-diffusion visualization technique arises from its automatic density distribution. This benefit suggests using the technique to augment other vector visualization techniques. We demonstrate this utility by augmenting a LIC visualization with a reaction-diffusion visualization. Finally, the reaction-diffusion visualization method provides a technique that can be used for streamline and glyph placement.