Spiral structures and spectra in two-dimensional turbulence

Abstract

Saffman argues that in decaying two-dimensional turbulence approximate dis-continuities of vorticity will form, and the energy spectrum will fall off as k −4 Saffman assumes that these discontinuities are well separated; in this paper, we examine how accumulation points of such discontinuities may give an energy spectrum of between k −4 and k −3 . In particular we examine the energy spectra of spiral structures which form round the coherent vortices that are observed in numerical simulations of decaying two-dimensional turbulence. If the filaments of the spiral are assumed to be passively advected, the instantaneous energy spectrum has a $k^{-11/3}$ range. Thus we come some way to reconciling the argument of Saffman and the k −3 energy spectrum predicted by models of quasi-equilibrium two-dimensional turbulence based on a cascade of enstrophy in Fourier space.