Velocity, energy, and helicity of vortex knots and unknots

Maggioni, Francesca; Alamri, Sultan Z.; Barenghi, Carlo F.; Ricca, Renzo L.

doi:10.1103/physreve.82.026309

Cited by 40 publications

(48 citation statements)

References 39 publications

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“…Moreover, since for w > 1 and λ small, C is generally small (Figure 4.4a), the main contribution to the drift velocity is given by the leading order term (eq. (4.4)), in good agreement with [30], where small-amplitude vortex torus knots/unknots with w > 1 are found to move essentially as fast as the reference vortex ring of same size and vorticity. …”

Section: Asymptotic Formula and Leading Order Termssupporting

confidence: 75%

“…The translational velocity and kinetic energy of vortex torus knots/unknots, calculated by numerical integration of the Biot-Savart law, are related in [30] to knot complexity, given by the winding number. Here we have derived for the first time by analytical means an integral formula for higher-order terms of the binormal component of the self-induced velocity at a given point, and we have confirmed the importance of global geometric contributions by showing the dominance of non-local terms.…”

Section: Asymptotic Formula and Leading Order Termsmentioning

confidence: 99%

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Induction effects of torus knots and unknots

Oberti¹,

Ricca²

2017

J. Phys. A: Math. Theor.

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The induction effects due to a steady source field in the shape of a torus knot or unknot filament are analysed in detail. Similar studies for rectilinear, circular or helical geometries have been done in the past, but very little is known for more complex geometries and topologies. Torus knots provide a rare example of closed, space curves of non-trivial topology, that admit a mathematically simple description; for this reason they represent an interesting case study to consider. Moreover, since torus knots are also a good mathematical model for studying braided field line structures, the present work provides useful information for a wide range of possible applications, from physical sciences (solar physics and astrophysics, vortex dynamics, fusion physics) to technology (telecommunication, new materials design, data analysis). The work is organized in 4 chapters.In chapter 1 we present a comprehensive study of geometric and topological properties of torus knots and unknots. By using a standard parametrization, we demonstrate the existence, and determine the location, of inflection points for a given critical configuration, and prescribe the condition for removing the singularity associated with torsion at the inflection point. We show that, to first approximation, total length grows linearly with the number of coils, and it is proportional to the minimum crossing number of the knot type. By taking the winding number, given by the ratio between meridian and longitudinal wraps, as measure of topological complexity of the knot, we analyse its influence on several global quantities, such as total length, curvature, torsion and writhe.In chapter 2 we analyse the influence of the winding number and other geometric properties on induction, energy and helicity. This is done by assuming the physical filament of infinitesimally small cross-section and by using the Biot-Savart law adapted for the particular parametrization chosen. Field line patterns of the induced field are obtained for a large family of knots/unknots on several crosssectional planes. The intensity of the induced field is shown to depend linearly on the number of toroidal coils. We provide bounds on energy, and an estimate of helicity in terms of writhe.In chapter 3 we compare local and global induction contributions in relation to the winding number, by providing asymptotic expansions of the integrand function. We show that in general local leading order terms are not sufficient to provide accurate global information; nevertheless, for some values of the winding number local and global behaviours are found to be in good agreement.In chapter 4 we investigate the influence of the winding number on the binormal component of the self-induction a point asymptotically near to the source field. Since in the limit the Biot-Savart integral becomes singular, we apply the analytical prescription of Moore and Saffman (1972) to regularize it. While to leading order the self-induction is proportional to local curvature, we derive an integral formula for nex...

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Section: Asymptotic Formula and Leading Order Termssupporting

confidence: 75%

Section: Asymptotic Formula and Leading Order Termsmentioning

confidence: 99%

Induction effects of torus knots and unknots

Oberti¹,

Ricca²

2017

J. Phys. A: Math. Theor.

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“…5b, and suppose that this results from the projection of a poloidal coil in space. Since the central area has index þ1 and the external lobes have all indices À1, by (8) [14], and confirmed by more recent work by Maggioni et al [15].…”

Section: Linear and Angular Momentum Of A Vortex Tangle From Geometrisupporting

confidence: 58%

Structural Complexity of Vortex Flows by Diagram Analysis and Knot Polynomials

Ricca

2013

How Nature Works

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In this paper I present and discuss with examples new techniques based on the use of geometric and topological information to quantify dynamical information and determine new relationships between structural complexity and dynamical properties of vortex flows. New means to determine linear and angular momenta from standard diagram analysis of vortex tangles are provided, and the Jones polynomial, derived from the skein relations of knot theory is introduced as a new knot invariant of topological fluid mechanics. For illustration several explicit computations are carried out for elementary vortex configurations. These new techniques are discussed in the context of ideal fluid flows, but they can be equally applied in the case of dissipative systems, where vortex topology is no longer conserved. In this case, a direct implementation of adaptive methods in a real-time diagnostics of real vortex dynamics may offer a new, powerful tool to analyze energy-complexity relations and estimate energy transfers in highly turbulent flows. These methods have general validity, and they can be used in many systems that display a similar degree of self-organization and adaptivity.

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“…To shed light onto this problem, energy, motion and stability of vortex knots have been examined theoretically and numerically using the classical theory of thincored vortex filaments. In this approach, the governing incompressible Euler dynamics is expressed by the Biot-Savart law or by its local induction approximation (LIA) [12,13].…”

Section: Introductionmentioning

confidence: 99%

Vortex knots in a Bose-Einstein condensate

2012

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We present a method for numerically building a vortex knot state in the superfluid wave-function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and stability of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how unstable vortex knots break up into vortex rings.

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Velocity, energy, and helicity of vortex knots and unknots

Cited by 40 publications

References 39 publications

Induction effects of torus knots and unknots

Induction effects of torus knots and unknots

Structural Complexity of Vortex Flows by Diagram Analysis and Knot Polynomials

Vortex knots in a Bose-Einstein condensate

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