By Alexander John Taylor
In this thesis, the writer develops numerical recommendations for monitoring and characterising the convoluted nodal strains in three-d area, analysing their geometry at the small scale, in addition to their worldwide fractality and topological complexity---including knotting---on the big scale. The paintings is very visible, and illustrated with many appealing diagrams revealing this unanticipated point of the physics of waves. Linear superpositions of waves create interference styles, this means that in a few locations they enhance each other, whereas in others they thoroughly cancel one another out. This latter phenomenon happens on 'vortex traces' in 3 dimensions. more often than not wave superpositions modelling e.g. chaotic hollow space modes, those vortex traces shape dense tangles that experience by no means been visualised at the huge scale sooner than, and can't be analysed mathematically through any recognized suggestions.
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Extra info for Analysis of Quantised Vortex Tangle
3 that they are appropriately similar to the isotropic RWM, via comparison of their statistics with known analytic results on the isotropic limit. 7 Variations on Random Waves 33 (c) intensity (a) phase (b) Fig. 17 A random degenerate energy 75 eigenfunction of the 3-torus. a and b show the intensity and phase respectively in a two-dimensional slice through the 3-torus, demonstrating a typical vortex pattern, while c shows vortex lines in three dimensions; each vortex is given a distinct colour, and much of the total vortex arclength is in the single green example.
7 Variations on Random Waves 37 (c) intensity (a) phase (b) Fig. 19 Random degenerate eigenfunctions of the 2-sphere and 3-sphere. a and b depict the intensity and complex phase respectively in a random superposition of degenerate energy 20 spherical harmonics of the 2-sphere, a pattern that will match (up to a statistical scaling) the pattern on a great-sphere cross section of the 3-sphere. 27), via stereographic projection; vortices are cut off where they pass close to ψ = π , and appear much denser in the centre where ψ is near to 0.
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. This has implications for certain statistics, discussed further in Sect. 1. 7 Variations on Random Waves 37 (c) intensity (a) phase (b) Fig. 19 Random degenerate eigenfunctions of the 2-sphere and 3-sphere.
Analysis of Quantised Vortex Tangle by Alexander John Taylor