Analysis of Quantised Vortex Tangle by Alexander John Taylor

By Alexander John Taylor

In this thesis, the writer develops numerical strategies for monitoring and characterising the convoluted nodal strains in 3-dimensional house, analysing their geometry at the small scale, in addition to their international fractality and topological complexity---including knotting---on the big scale. The paintings is extremely visible, and illustrated with many attractive diagrams revealing this unanticipated element of the physics of waves. Linear superpositions of waves create interference styles, this means that in a few locations they increase each other, whereas in others they thoroughly cancel one another out. This latter phenomenon happens on 'vortex strains' in 3 dimensions. normally wave superpositions modelling e.g. chaotic hollow space modes, those vortex strains shape dense tangles that experience by no means been visualised at the huge scale prior to, and can't be analysed mathematically by way of any identified thoughts.

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All images show a square region of side length 7 λ Many of these statistical results on vortex lines are obtained from integration on the Gaussian random wave model, in which the values of field quantities are written in terms of correlated Gaussian random variables which can be linearised to an independent integral [19]. e. ξ, ξx , ξx y etc. where the subscripts indicate differentials) since these quantities are not independent; in practice the calculation can be simplified by linearising their matrix of correlations to obtain the same number of now-independent Gaussian-distributed quantities.

Unlike some of the models above, the lengths of the polygon edges are not uniform but instead take a beta distribution [80], though this does not seem to significantly affect statistics. 6 Other Random Tangles and Random Walks 29 Fig. 14 Three example closed random walks. 2. a, b and c have lengths of 100, 1000 and 10000 segments respectively, each coloured along their length only to distinguish different regions of the curves the sum of edge segments happens to close, and selecting the starting position from within these regions.

Although it is not necessarily clear in Fig. 19c, the vortex pattern includes an inversion symmetry of the eigenfunctions under the rotation ψ → π −ψ, θ → π −θ and φ → φ + π . 27), Ylm (π − θ, φ + π ) = N −l l+1 C N −l (cos ψ), so Y Nlm (π − ψ, π − (−1)l Yl and C l+1 N −l (cos(ψ + π )) = (−1) N θ, φ + π ) = (−1) Y Nlm (ψ, θ, φ). The nodal line structure is invariant to this parity and so is simply symmetric under this rotation; any vortex structure appears twice in any degenerate eigenfunction.

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