Time-dependent geometry and energy distribution in a spiral vortex layer.

The purpose of this paper is to study how the geometry and the spatial distribution of energy fluctuations of different length scales in a spiral vortex layer are related to each other in a time-dependent way. The numerical solution of Krasny [J. Comput. Phys. 65, 292 (1986)], corresponding to the development of the Kelvin-Helmholtz instability, is analyzed in order to determine some geometrical features necessary for the analysis of Lundgren's unstrained spiral vortex. The energy distribution of the asymptotic solution of Lundgren characterized by a similar geometry is investigated analytically (1) in the wavelet radius-scale space, with a wavelet selective in the radial direction, and (2) in the wavelet azimuth-scale space, with a wavelet selective in the azimuthal direction. Energy in the wavelet radius-scale space is organized in "blobs" distributed in a way determined by the Kolmogorov capacity of the spiral D(K) in [1,2] (which determines the rate of accumulation of spiral turns). As time evolves these blobs move towards the small scale region of the wavelet radius-scale space, until their scale is of the order of the diffusive length scale square root[nut], where t is the time and nu is the kinematic viscosity. In contrast, energy in the wavelet azimuth-scale space is not localized, and is characterized by a shear-augmented viscous cutoff proportional to sqaure root[nut(3)]. An accelerated viscous dissipation of the enstrophy and energy of Lundgren's spiral vortex is found for D(K) > 1.75, but not for D(K) < or =1.75.