Nonlinearity depletion, elementary excitations and impulse formulation in vortex dynamics

In this article, a method for detecting depletion of non-linearity is proposed on the basis of a comparison between continuous and discrete dynamics in the impulse formulation of vortex dynamics. The idea is: if a flow field is represented by a collection of tiny vortex rings, the dynamics may be described well by the ODEs, if they are well separated and do not undergo close interaction. Otherwise, significant deformation of core should take place leading to depletion. This is tested against a system of six dipoles approaching the origin which models the late stage of the Kida-Pelz vortex, a candidate for a blowup in the 3D Euler equations. The ODEs suggest that a singularity should occur off the origin when the self-interaction is taken into account. Simulations of fluid equations show that locations of the dipoles are well described by the ODEs up to the time when the mutual distances are comparable with their sizes. It is pointed out that the crucial difference between continuous and discrete systems is the presence or absence of a fixed characteristic length scale. A method with higher-order terms of the multi-pole expansions is suggested for a more quantitative and systematic detection of depletion.