2015/05/31 by Darryl D. Holm, Henry O. Jacobs
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Euler equations #Fluid dynamics #Gravitational singularity #Inviscid flow #Lattice Boltzmann Simulation Studies #Orbital Angular Momentum in Optics #Regularization (linguistics) #Symplectic geometry #Vortex #Vortex stretching #Vorticity #math.DS #math.SG #physics.flu-dyn
paper · pdf · doi:10.1007/s00332-017-9367-4
Published in J Nonlinear Sci
openalex created_date 2016/06/24 · openalex publication_date 2017/03/16 · arxiv created 2018/10/20 · arxiv updated 2018/10/23 · openalex updated_date 2026/08/05
Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.