Vorticity statistics of bounded two-dimensional turbulence

Vorticity statistics play an important role in the determination of small-scale dynamics in forced two-dimensional turbulence. On the basis of the Hölder-continuity of the vorticity field ømega(t,\bfx), the scaling behavior of vorticity structure functions S_p(ømega(\ell)), of order p, provides clues on small-scale intermittency. Confirming earlier ideas of Sulem and Frisch (JFM 72, 1975), Eyink (Phys. D 91, 1996) proved the following scaling of the second-order structure function S_2(ømega(\ell))\equiv\langle|ømega(t,\bfx+\bfr)-ømega(t,\bfx)|^2\rangle\sim\ell^\zeta_2, with \quad\zeta_2\leq 2/3 and \quad\ell\leq\ell_f. Here, \ell=|\bfr|, \ell_f is the typical energy-injection scale, associated to an external forcing and the brackets \langle\cdot\rangle denote combined space- and time-averaging. The only assumption used to derive this scaling was a constant enstrophy flux to small scales, in the so-called enstrophy cascade range. On the contrary, using the classical Batchelor argument for the advection of a passive scalar, Falkovich and Lebedev (PRE 50, 1994) argued that one must have \zeta_p=0 for all p. With new direct numerical simulations we address these issues for a bounded square domain, using the no-slip boundary condition for the velocity. Our results are compared with the earlier experimental results of Paret and Tabeling (PRL 83, 1999).