Renzo L. Ricca

Helicity and the Calugareanu invariant

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics

In this paper we present for the first time a detailed account of the work of L.S. Da Rios and T. Levi-Civita on what is believed to be one of the first major contributions to three-dimensional vortex filament dynamics. Their work spanned a period of almost 30 years, from 1906 to 1933, and despite many publications remained almost unnoticed throughout this century. After a partial re-discovery (Ricca, 1991a), new material has now been found and is presented here with a full review of their work in relation to the present state of the art in non-linear mechanics and vortex dynamics. Their results include the conception of the localized induction approximation (LIA) for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations and the stability analysis of circular vortex filaments. In the light of modern developments in non-linear fluid mechanics, their work strikes for modernity and depth of results. Even more striking is the fact that this work remained obscure for almost a century. The results of Da Rios are particularly important in the study of integrable one-dimensional systems and vortex filament motion; Levi-Civitas work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.

The effect of torsion on the motion of a helical vortex filament

In this paper we analyse in detail, and for the first time, the role of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement of the vortex filament in the fluid, is closely analysed. The study is performed by applying the prescription of Moore & Saffman (1972) to helices of any pitch and a new asymptotic integral formula is derived. We give a rigorous proof that the Kelvin regime and its limit behaviour are obtained as a limit form of that integral asymptotic formula. The results are compared with new calculations based on the re-elaboration of Hardins (1982) approach and with results obtained by Levy & Forsdyke (1928) and Widnall (1972) for helices of small pitch, here also re-elaborated for the purpose.

Topological Ideas and Fluid Mechanics

The use of topological ideas in physics and fluid mechanics dates back to the very origin of topology as an independent science. In a brief note in 1833 Karl Gauss, while lamenting the lack of progress in the “geometry of position” (or Geometria Situs, as topology was then known I, gives a remarkable example of the relationship between topology and measurable physical quantities such as electric currents. He considers two inseparably linked circuits, each of them a copper wire with ends joined, and flowing electric current. Without comment he puts forward a formula that gives the relationship between the magnetic action induced by the currents and a pure number that depends only on the type of link, and not on the geometry. This number is a topological invariant now known as the linking number. The formula, as well as the very first studies in topology done by Johann Benedict Listing in 1847, became known to Kelvin (then William Thomson), James Clerk Maxwell and Peter Guthrie Tait in Britain.

Evolution of vortex knots

For the first time since Lord Kelvins original conjectures of 1875 we address and study the time evolution of vortex knots in the context of the Euler equations. The vortex knot is given by a thin vortex filament in the shape of a torus knot F p,q (p > 1, q > 1; p, q co-prime integers). The time evolution is studied numerically by using the Biot-Savart (BS) induction law and the localized induction approximation (LIA) equation. Results obtained using the two methods are compared to each other and to the analytic stability analysis of Ricca. The most interesting finding is that thin vortex knots which are unstable under the LIA have a greatly extended lifetime when the BS law is used. These results provide useful information for modelling complex structures by using elementary vortex knots

How tangled is a tangle

Abstract New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments. The tangle is produced by standard numerical simulation of superfluid helium turbulence, which we use as a benchmark for numerical investigation of complex systems. We find that the measures used, based on crossing number information, are good indicators of generic behaviour and detect accurately a tangle’s complexity. Direct measurements of kinetic helicity are found to be in agreement with the other complexity-based measures, proving that helicity is also a good indicator of structural complexity. We find that complexity-based measure growth rates are consistently similar to one another. The growth rate of kinetic helicity is found to be twice that of energy.

An introduction to the geometry and topology of fluid flows

Preface. Photograph of H.K. Moffatt. I: Eight Problems for the XXI Century. Some Remarks on Topological Fluid Mechanics H.K. Moffatt. II: Mathematics Background. Differential Geometry of Curves and Surfaces R. Langevin. Topology in Four Days T. Tokieda. Elements of Classical Knot Theory C. Weber. An Introduction to Knot Theory L.H. Kauffman. Fluid Mechanics and Mathematical Structures P. Boyland. III: Geometry and Topology of Fluid Flows. Introduction to a Geometrical Theory of Fluid Flows and Dynamical Systems T. Kambe. Streamline Patterns and their Bifurcations Using Methods from Dynamical Systems M. Brons. Topological Features of Inviscid Flows R. Grhist, R. Komendarczyk. Geometric and Topological Aspects of Vortex Motion R.L. Ricca. Topology Bounds the Energy B.A. Khesin. Measures of Topological Structure in Magnetic Fields M.A. Berger. Diffeomorphisms, Braids and Flows A. Shnirelman. Variational Principles, Geometry and Topology of Lagrangian-Averaged Fluid Dynamics D.D. Holm. IV: Reconnections and Singularities. The Geometry of Reconnection G. Hornig. Euler Singularities from the Lagrangian Viewpoint S. Childress. Analysis of a Candidate Flow for Hydrodynamic Blowup R.B. Pelz. Subject Index.

The Helicity of a Knotted Vortex Filament

The helicity H associated with a knotted vortex filament is considered. The filament is first constructed starting from a circular tube, in three stages involving injection of (integer) twist, deformation and switching of crossings. This produces a vortex tube in the form of an arbitrary knot K; each vortex line in the tube is a (trivial) satellite of K, and the linking number of any pair of vortex lines in the tube is the same integer n. It is shown that in these circumstances the helicity is given by H = nк 2 where к is the circulation associated with the tube. This result is discussed in relation to earlier works, in particular the work of Călugăreanu (1959, 1961) which establishes that, for a twisted ribbon with axis C the number n is the sum of three ingredients:

Physical interpretation of certain invariants for vortex filament motion under LIA

\frac{H}{{{\kappa ^2}}} = n = W(C) + T(C) + \frac{1}{{2\pi }}{[\Delta \Theta ]_C}