On long-lived vortices in 2-D viscous flows, most probable states of inviscid 2-D flows and a soliton equation

Abstract

This paper considers, in contraposition to the most probable states of quasi-inviscid theories, the status of the vorticity(ω)–streamfunction(ψ) relations satisfied by the long-lived vortices observed in some numerical simulations of decaying two-dimensional turbulence and in experiments in stratified fluids. For the case ω=−Δψ=sinh ψ, the circular solutions can be expressed in terms of the IIIrd Painlevé transcendent and dipolar solutions can be constructed by means of a Bäcklund transform.