Three-dimensional unsteady axisymmetric vortex solutions to the Bellamy-Knights equation and the distribution of boundary conditions

The Bellamy-Knights equation gives three-dimensional unsteady axisymmetric incompressible vortex solutions that are disconnected from the known exact steady solutions. In this paper, by introducing a new parameter that allows a change of the time scale, the Bellamy-Knights equation is transformed into an integro-differential equation that continuously connects with the equation for steady vortices. The new equation is viewed as describing the one-dimensional motion of a particle under conserved and nonconserved forces and leads to new regular solutions, in which the particle initially undergoes almost sinusoidal oscillations around the minimum of potential followed by a very rapid approach to the extrema of the potential. The regular solutions distribute discretely in the space of boundary conditions and are classified into types that are specified by the pattern of sequence of small and large oscillations of the axial component of velocity. Extending the solutions to singular ones, the boundary conditions for solutions of the same type are shown to form a continuous one-dimensional set, which we call a branch. The set of branches forms dense self-similar patterns with accumulation points, some of which coalesce to a single point. In an arbitrary neighborhood of the accumulation or coalescent point, any two distinct boundary conditions whose distance in the space of boundary conditions is arbitrarily small give mutually distinctive solutions with a finite difference at some region of the spatial distance. This feature of the equations signifies the extreme sensitivity of inflowing vortex solutions to boundary conditions.