Search

The equations governing the evolution of quantum vortex defects subject to twist are derived in standard hydrodynamic form. Vortex defects emerge as solutions of the Gross-Pitaevskii equation, that by Madelung transformation admits a hydrodynamic description. Here, we consider a vortex defect subject to superposed twist due to the rotation of the phase of the wave function. We prove that, when twist is present, the corresponding Hamiltonian is non-Hermitian and determine the effect of twist on the energy expectation value of the system. We show how twist diffusion may trigger linear instability, a property directly related to the non-Hermiticity of the Hamiltonian. We derive the correct continuity equation and, by applying defect theory, we obtain the correct momentum equation. Finally, by coupling twist kinematics and vortex dynamics we determine the full set of hydrodynamic equations governing quantum vortex evolution subject to twist.

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. Results have been obtained for 56 knots and unknots up to 51 crossings. In agreement with previous numerical results we prove that in general the propagation speed increases with the number of toroidal coils, but we notice that the number of poloidal coils may greatly modify the motion. Indeed we prove that for increasing aspect ratio and number of poloidal coils vortex motion can be even reversed, in agreement with previous numerical observations. These results demonstrate the importance of three-dimensional features in vortex dynamics and find useful applications to understand helicity and energy transfers across scales in vortical flows.

Here we show how to apply a recently introduced method based on the geometric interpretation of linear momentum of vortex lines to determine dynamical properties of a network of knots and links. To show how the method works and to prove its feasibility, we consider the evolution of quantum vortices governed by the Gross-Pitaevskii equation. Accurate estimates of the momentum of interacting and reconnecting vortex rings, links, and knots are determined. The method is of general validity and it proves particularly useful in practical situations where no analytical information is available. It can be easily adapted to situations where morphological information can be extracted from experimental or computational data, thus providing a powerful tool for real-time diagnostics of vortex filaments or other networks of filamentary structures.

Topological fluid mechanics is a crucial area of classical mechanics, with remarkable theoretical significance and wide applications in practice. In this paper, we expect to present an introductory review of the direction, for the purpose of attracting more domestic researchers of China to enter into this important area. Our emphasis is placed on the topological essence of fluid helicity. We will give the relationship between helicity and mathematical knotted field theory (i.e., that between helicity and the mutual- and self-linking numbers, as well as the fluid knot polynomial topological invariants constructed in terms of helicity recently discovered by the authors), and give an introduction to the international research on energy-structural complexity relationship for a fluid vortex knot ensemble. Moreover, typical numerical methods are also demonstrated through examples of reconnections occurring in superfluid vortex knots/links. Expectedly, a combination of the theoretical framework and numerical techniques may contribute to the reader a comprehensive understanding of the main target, research methodology and potential technical difficulties in practice in this interdisciplinary field of cutting-edge research world-wide.

In this paper we show that twist, defined in terms of rotation of the phase associated with quantum vortices and other physical defects effectively deprived of internal structure, is a property that has observable effects in terms of induced axial flow. For this we consider quantum vortices governed by the Gross–Pitaevskii equation (GPE) and perform a number of test cases to investigate and compare the effects of twist in two different contexts: (i) when this is artificially superimposed on an initially untwisted vortex ring; (ii) when it is naturally produced on the ring by the simultaneous presence of a central straight vortex. In the first case large amplitude perturbations quickly develop, generated by the unnatural setting of the initial condition that is not an analytical solution of the GPE. In the second case much milder perturbations emerge, signature of a genuine physical process. This scenario is confirmed by other test cases performed at higher twist values. Since the second setting corresponds to essential linking, these results provide new evidence of the influence of topology on physics.

By numerically solving the three-dimensional Gross-Pitaevskii equation we analyze the cascade process associated with the evolution and decay of a pair of linked vortex rings. The system decays through a series of reconnections to produce finally three unlinked, unfolded, almost planar vortex loops. Total helicity, initially zero, remains unchanged throughout the process. The gradual transfer from writhe (due to initial linking) to twist helicity, followed by a continuous relaxation of twist across scales during the evolution is shown to be a generic mechanism that consistently takes place on each individual component.

The process of vortex cascade through continuous reduction of topological complexity by stepwise unlinking, that has been observed experimentally in the production of vortex knots (Kleckner & Irvine, 2013), is shown to be reproduced in the branching of the skein relations of knot polynomials (Liu & Ricca, 2015) used to identify topological complexity of vortex systems. This observation can be usefully exploited for predictions of energy-complexity estimates for fluid flows.

Here we show that under quantum reconnection, simulated by using the three-dimensional Gross-Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that the total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion, and intrinsic twist of the reconnecting vortex rings.

In this paper we derive and compare numerical sequences obtained by adapted polynomials such as HOMFLYPT, Jones and Alexander-Conway for the topological cascade of vortex torus knots and links that progressively untie by a single reconnection event at a time. Two cases are considered: the alternate sequence of knots and co-oriented links (with positive crossings) and the sequence of two-component links with oppositely oriented components (negative crossings). New recurrence equations are derived and sequences of numerical values are computed. In all cases the adapted HOMFLYPT polynomial proves to be the best quantifier for the topological cascade of torus knots and links.

In this paper we analyse in detail, and for the first time, the rôle 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 régime 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 Hardin's (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.

A geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in Euler flows. For thin filaments under the so-called localized induction approximation the components of linear momentum admit interpretation in terms of projected area. By computing the signed areas of the projected graph diagrams associated with the vortex tangle, we show how to calculate the two momenta of the system by complexity analysis of tangle diagrams. This method represents a novel technique to extract dynamical information of complex systems from geometric and topological properties and provides a potentially useful tool to test the accuracy of numerical methods and investigate scale distribution of fluid dynamical properties of vortex flows.

In this paper we introduce and discuss new concepts useful to analyse and characterize patterns of vortex lines in fluid flows. We define measures of tropicity to identify ‘tubeness’, ‘sheetness’ and ‘bulkiness’ of vortex lines and to measure the spreading of field lines about preferred directions. Algebraic, geometric and topological measures based on crossing number information are discussed and are put in relation to the kinetic helicity and the energy of the fluid system.

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-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.

Geometric and topological aspects associated with integrability of vortex filament motion in the Localized Induction Approximation (LIA) context (which includes a family of local dynamical laws) are discussed. We show how to interpret integrability in relation to the Biot-Savart law and how soliton invariants can be interpreted in terms of global geometric functionals of knotted solutions. Under the basic (zeroth-order) LIA, we prove that vortex filaments in the shape of torus knots T p, q (p, q co-prime) with (q/p)>1 are stable, whereas those with (q/p)

We present here the natural algebraic-geometrical generalization of the Da Rios-Betchov intrinsic equations governing curvature and torsion of an isolated vortex string moving in an unbounded, perfect fluid flow. The filament is embedded in a manifold that can be assumed to be homeomorphic to an odd-dimensional Euclidean space, and whose connection we do not assume to be torsionfree. We suggest how to account for fluid compressibility in the ambient space by its geometrization, and we discuss some special cases of physical interest such as the torsion-free affine connection case and the Riemannian connection case. Finally, we point out the role our results might have in the context of soliton studies.

For the first time since Lord Kelvin's 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 [Tscr ]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 (1993, 1995). 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.