Vortex models based on similarity solutions of the two-dimensional diffusion equation

Abstract

In this paper, a class of two-dimensional (2D) vortex models is analyzed, which is based on similarity solutions of the diffusion equation. If the nonlinear advective term is neglected, the 2D Navier-Stokes equation reduces to a linear problem, for which a complete orthonormal set of eigenfunctions is known on an unbounded 2D domain. Some of the basic modes represent models for diffusing monopoles, dipoles, and tripolar vortices, which evolve self-similarly in time. Here, we mainly confine ourselves to an analysis of the dipole solution. In several respects, especially the decay and, to a lesser extent, the lateral expansion properties, the dipole model appears to be in fair agreement with the real evolution of dipolar vortices for finite Reynolds number, as obtained from numerical simulations of the full 2D Navier-Stokes equations. However, the simulations reveal that nonlinear effects result in small differences compared to the evolution according to the model. The most important nonlinear effect that was observed is the formation of “tails” of vorticity in the wake of the dipole. After a while, any initial condition leads to a vorticity distribution lying in between the viscous similarity solution and the Lamb dipole solution, which represents the limit of a stationary, inviscid flow. The exact form of the vorticity distribution is believed to be determined by an equilibrium between diffusion of vorticity through the separatrix and advection of vorticity into the wake of the dipole, which results in the formation of vorticity tails. A comparison revealed profound qualitative agreements between the model together with the simulations and dipolar vortex structures that were studied by laboratory experiments in stratified fluids.