Multipole Vortex Blob (MVB): Symplectic Geometry and Dynamics

TL;DR

This paper introduces a new singular vortex theory for regularized Euler equations, allowing for stronger vorticity singularities and revealing complex multiscale dynamics through symplectic geometry and Hamiltonian characterization.

Contribution

It develops a framework for vortex dynamics involving derivatives of delta functions, extending traditional vortex blob methods and exploring their symplectic structure.

Findings

Vorticity can be modeled as derivatives of delta functions in regularized Euler equations.

The symplectic geometry of these vortex structures is characterized as Hamiltonian.

The approach enables capturing multiscale phenomena below the regularization length scale.

Abstract

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 to 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.