Self-similarity of decaying two-dimensional turbulence

Simulations of decaying two-dimensional turbulence suggest that the one-point vorticity density has the self-similar form P omega similar to t f(omega t) implied by Batchelor's (1969) similarity hypothesis, except in the tails. Specifically, similarity holds for \omega\ < omega(m), while p(omega) falls off rapidly above. The upper bound of the similarity range, omega(m), is also nearly conserved in high-Reynolds-number hyperviscosity simulations and appears to be related to the average amplitude of the most intense vortices (McWilliams 1990), which was an important ingredient in the vortex scaling theory of Carnevale et al. (1991). The universal function f also appears to be hyperbolic, i.e. f(x) similar to c/2\X\(1+qc), for \x\ > x*, where q(c) = 0.4 and x* = 70, which along with the truncated similarity form implies a phase transition in the vorticity moments [GRAPHICS] between the self-similar 'background sea' and the coherent vortices. Here c(q) and c are universal. Low-order moments are therefore consistent with Batchelor's similarity hypothesis whereas high-order moments are similar to those predicted by Carnevale et al. (1991). A self-similar but less well-founded expression for the energy spectrum is also proposed. It is also argued that omega(s) = x*/t represents 'mean sea-level', i.e. the (average) threshold separating the vortices and the sea, and that there is a spectrum of vortices with amplitudes in the range (omega(s),omega(m)). The total area occupied by vortices is also found to remain constant in time, with losses due to mergers of large-amplitude vortices being balanced by gains due to production of weak vortices. By contrast, the area occupied by vortices above a fixed threshold decays in time as observed by McWilliams (1990).