FINITE-TIME VORTEX SINGULARITY AND KOLMOGOROV SPECTRUM IN A SYMMETRICAL 3-DIMENSIONAL SPIRAL MODEL

A recent analytical model of three-dimensional Euler flows [Phys. Rev. Lett. 69, 2196 (1992)] which exhibits a finite-time vortex singularity is developed further. The initial state is symmetric and contains a velocity null (stagnation point) which is collinear with two vorticity nulls. Under some assumptions, it is shown by asymptotic analysis of the Euler equation that the vorticity blows up at the stagnation point as inverse time in a locally self-similar manner. The spatial structure of the inviscid flow in the vicinity of the singularity involves disparate small scales. The effect of a small but finite viscosity is shown to arrest the formation of the singularity. The presence of spiral structure in the initial conditions leads naturally to the model developed by Lundgren [Phys. Fluids 25, 2193 (1982)] in which the gradual tightening of spirals by differential rotation provides a mechanism for transfer of energy to small spatial scales. It is shown by asymptotic analysis of the Navier-Stokes equation, that a time-average over the lifetime of the spiral vortex in the present model yields the Kolmogorov spectrum.