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 dist ribution of energy fluctuations of different length scales in a spiral vort ex layer are related to each other in a time-dependent way. The numerical s olution of Krasny [J. Comput. Phys. 65, 292 (1986)], corresponding to the d evelopment of the Kelvin-Helmholtz instability, is analyzed in order to det ermine some geometrical features necessary for the analysis of Lundgren's u nstrained spiral vortex. The energy distribution of the asymptotic solution of Lundgren characterized by a similar geometry is investigated analytical ly (1) in the wavelet radius-scale space, with a wavelet selective in the r adial direction, and (2) in the wavelet azimuth-scale space, with a wavelet selective in the azimuthal direction. Energy in the wavelet radius-scale s pace is organized in "blobs" distributed in a way determined by the Kolmogo rov capacity of the spiral D-K is an element of [1,2] (which determines the rate of accumulation of spiral turns). As time evolves these blobs move to wards the small scale region of the wavelet radius-scale space, until their scale is of the order of the diffusive length scale root vt, where t is th e time and v is the kinematic viscosity. In contrast, energy in the wavelet azimuth-scale space is not localized, and is characterized by a shear-augm ented viscous cutoff proportional to root vt(3). An accelerated viscous dis sipation of the enstrophy and energy of Lundgren's spiral vortex is found f or D-K>1.75, but not for D(K)less than or equal to 1.75. [S1063-651X(99)006 05-4].