ENERGY, HELICITY AND CROSSING NUMBER RELATIONS FOR COMPLEX FLOWS

Algebraic and topological measures based on crossing number relations
provide bounds on energy and helicity of ideal fluid flows and can be
used to quantify morphological complexity of complex tangles of vortex
filaments [1,2]. Recent work [3], based on simulation of superfluid
vortex tangles, demonstrates that structural complexity can indeed be
identified with crossing number measurements, and provides evidence for
possible new relations between complexity and energy of structured flows.
These results find useful applications, from diagnostics of turbulent
flows (superfluid and classical) to energy estimates of
complex coherent structures.


[1] Ricca, R.L. 2002 Energy, helicity and crossing number relations
for complex flows. In Tubes, Sheets and Singularities in Fluid
Dynamics (ed. K. Bajer & H.K. Moffatt), pp. 225-230.
NATO ASI Series II, to appear, Kluwer.
[2] Ricca, R.L. 2001 Geometric and topological aspects of vortex
motion. In An Introduction to the Geometry and Topology of Fluid
Flows (ed. R.L. Ricca), pp. 203-228. NATO ASI Series II, vol. 47,
Kluwer.
[3] Barenghi, C.F., Ricca, R.L. & Samuels, D.C. 2001 How tangled is a
tangle? Physica D vol. 157, 197-206.