Decaying, two-dimensional, Navier-Stokes turbulence at very long times

Abstract

Two-dimensional Navier-Stokes turbulent decay has been followed numerically for very long times. The code is spectral, satisfies periodic boundary conditions, and does not make use of hyperviscosity or any small-scale smoothing. The resolution is (512) 2, the initial Reynolds number based on the box dimension is about 14000, and the run continues for over 190 initial eddy turnover times. Isolated vortices form, and the turbulence has become highly intermittent in the manner seen by McWilliams and Brachet et al., for times greater than about 30. However, it is not the case that merger of like-signed vortices stops; it only slows down. By t = 210, the final merger is complete, and the vorticity distribution is dominated by one large vortex of either sign. The negative (positive) vortices each occupy about two percent of the box area, and together account for over 98 percent of the total enstrophy. Their alignment suggests the formation of an Ewald lattice with a basic cell containing two point vortices. The ratio of enstrophy to energy continues to decrease monotonically, and the picture is consistent with a “selective decay” process, as described some time ago. A not-entirely-understood phenomenon is the concentration of the vorticity into two cores, suggestive of a negative-temperature state of the discrete line vortex model.