On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid

Abstract

A wide class of solutions of the steady Euler equations, representing localized rotational disturbances imbedded in a uniform stream U0 is inferred by considering the process of magnetic relaxation to analogous magnetostatic equilibria. These solutions, which may be regarded as generalizations of vortex rings, are characterized by their streamline topology, distinct topologies giving rise to distinct solutions.Particular attention is paid to the class of axisymmetric solutions described by Stokes stream function ψ(s, z). It is argued that the appropriate topological ‘invariant’ characterizing the flow is the function Vψ representing the volume inside toroidal surfaces ψ = const, in the region of closed streamlines where ψ > 0. This function is described as the ‘signature’ of the flow, and it is shown that in a certain sense, flows with different signatures are topologically distinct. The approach yields a method by which flows of arbitrary signature V(ψ) may in principle be found, and the corresponding vorticity ωφ = sFψ calculated.