Introduction to Focus Issue: Objective Detection of Coherent Structures.

Abstract

Coherent structures are persistent localized features in time-varying fields. Some examples of physical fields in which coherence is of interest include pressure, temperature, and density of a deforming continuum, such as water, air, a granular material or a solid body. A first-order approximation to coherence in any (generally diffusive) physical field is coherence in the deformation of the carrier material. To date, a number of mathematical and ad-hoc methods have been developed, all striving to highlight coherent material regions, i.e., temporally coherent sets formed by trajectories of material particles in the phase space. The practitioner, however, is hard pressed to decide which of these methods is suitable for investigating a yet unknown flow. This is a crucial question in real-world applications, such as real-time forecasting and now-casting for environmental decisionmaking and control. A simple, yet universal, principle narrows down the panoply of methods significantly to those that are at least consistent with their stated goal of detecting material (or Lagrangian) structures. This principle is objectivity, i.e., invariance under changes of the observer. 10,13,14 This means that a coherent material set identified by an objective criterion in one observer frame should come out to be the same material set when the same criterion is applied in any other observer frame. For instance, sharply visible material coherent structures (fronts) block the transport of algae populations into certain regions of the ocean surface. 20 An observer on a cruising ship or another one on a circling airplane will visually identify the same material points in the front even though those material points will traverse on different paths in the frames of the two observes. Most coherent structures, however, are not so directly visible as the above example of a front. For instance, boundaries of coherent material eddies embracing and transporting volumes of ocean water with different salinity or temperature are notoriously difficult to identify. 4 The two observers mentioned above could then apply a material coherent structure detection method to the velocity field measured in their own frames. A third observer could also perform the same exercise from the shore (also in a rotating frame, given the motion of the earth). Clearly, if any of these observers reports a different assessment for the coherent material eddy boundary, the coherent structure method all three observers use is not self-consistent and hence is unreliable. More generally, objectivity is a necessary condition for the reliability of any coherent structure method. Any nonobjective method, even if it is an exact mathematical criterion, can at best be a sufficient condition, and hence can produce false negatives. Indeed, if the conditions of a nonobjective method for coherence are not satisfied in a chosen frame, they may still well be satisfied in another frame. The practitioner would have to check infinitely many different frames to exclude, with certainty, the possibility of a false negative signalled by the method in one frame. A nonobjective method without rigorous mathematics is even less helpful: it can produce both false positives and false negatives for material coherent structures.