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Texas A&M University

Events for 05/03/2023 from all calendars

Mathematics in Geosciences

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Time: 11:00AM - 12:00PM

Location: Bloc 302

Speaker: Paul Newton, University of Southern California

Title: Vortex lattices on the sphere in the large N limit

Abstract: Motivated by understanding vortex patterns on the sphere, such as von Karman vortex streets, we will describe a new method of calculating a broader class of lattice patterns on curved surfaces (I.e. sphere) that are free of discrete symmetries and can form in a random environment. We give an overview of an N-particle based Hamiltonian theory (derived from the Euler equations) which, if formulated in terms of the O(N2) interparticle distances, leads to the analysis of a nonnormal “configuration” matrix whose nullspace structure determines the existence or nonexistence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths, in principle, can be classified and calculated by a random-walk scheme which systematically uses the m smallest singular values as a ratchet mechanism to home in on lattices with m-dimensional nullspaces (any m). The resulting singular value distribution of the configuration matrix encodes detailed geometric properties of the lattice structures and allows us to compute the Shannon entropy of the lattice as a convenient measure of its disorder. We will show how the method works for relatively simple patterns (Platonic solids and von Karman vortex streets) and then focus on a computational Brownian ratchet method to calculate large N lattices that lack symmetry (P.K. Newton, G. Chamoun, SIAM Rev. Vol. 51 No. 3 501-542, 2009).