Self-similarity of decaying two-dimensional turbulence

Abstract

Simulations of decaying two-dimensional turbulence suggest that the one-point vorticity density has the self-similar form $P_\omega \sim t\;\;f(\omega t)$

implied by Batchelor's (1969) similarity hypothesis, except in the tails. Specifically, similarity holds for |ω| < ω_m, while p_ω falls off rapidly above. The upper bound of the similarity range, ω_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. $f(x) \sim c/2\vert x \vert^{1+q_c}$ for |x| > x*, where qc = 0.4 and x* = 70, which along with the truncated similarity form implies a phase transition in the vorticity moments $\langle \vert \omega\vert ^q\rangle \sim \left\{\begin{array}{ll} c_q t^{-q}, & -1 < q < q_c\cr c(q - q_c)^{-1} \omega _m^{q-q_c} t^{-q_c} & q > q_c, \end{array}\right.$ between the self-similar 'background sea' and the coherent vortices. Here Cq 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 ω_c = 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 (ω_s,ω_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 afixed threshold decays in time as observed by McWilliams (1990).