Towards a complexity measure theory for vortex tangles

In this paper we address the problem of measuring structural complexity of generic tangles of vortex lines in a fluid domain, by using a combination of geometric and topological techniques. To this end new concepts based on the idea of structural 'tropicity' are introduced to determine 'tubeness', "sheetness" and 'bulkiness' of a vortex tangle and to evaluate the degree of topological entanglement. A number of cases are considered: from highly organised, coherent vortex regions, given by the embedding of vortex coils, knots and links on nested tori, to less organised vortical flows, such as tangles of chaotic vortex lines. Various measures of linking (and helicity) are presented as well as estimates of writhing and crossing numbers based on geometric and topological information. Moreover, by using the concept of signature preserving flow we extend the definition of classical stability to include wilder vortex dynamics that during evolution preserve structural complexity. The tools and the new concepts presented in this paper are useful for the classification and study of general flow fields and can be employed to develop computational techniques for measuring structural complexity.