Merger and alignment in a reduced model for three-dimensional quasigeostrophic ellipsoidal vortices

Abstract

We investigate the interaction of two ellipsoidal vortices in the three-dimensional quasigeostrophic fluid equations by first studying a reduced model of vortex interaction, the ellipsoidal moment model, and second by comparing the results to corresponding numerical simulations. The ellipsoidal moment model approximates the interaction of two ellipsoidal lumps of potential vorticity by a finite-degree-of-freedom Hamiltonian system. This approximation is derived explicitly in the natural moment coordinate system to first order in the ratio of the size of the vortices to their separation. Using this Hamiltonian system for the case of initially spheroidal identical vortices, the linear stability of vertically aligned vortices is analyzed. A new dynamical criterion for vortex merger and alignment is proposed and shown to give a clear and reasonable boundary for vortex merger. A similar boundary is shown to exist in the size of the largest Lyapunov exponent, although not in the chaotic region as measured by the 0-1 test of Gottwald and Melbourne [Proc. R. Soc. London, Ser. A 460, 603 (2004)]. There is no such sharp boundary for vortex alignment in this reduced model. A series of numerical experiments confirms the accuracy of the merger criterion used in the ellipsoidal moment model. The numerical simulations also suggest a mechanism for understanding the process of vortex alignment in terms of vortex Rossby waves.