Hilbert statistics of vorticity scaling in two-dimensional turbulence

In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed ω(x, y) in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang transform, to a vorticity field obtained from a 8192² grid-points direct numerical simulation of the 2D turbulence with a forcing scale k_f = 100 and an Ekman friction. The measured joint probability density function p(C, k) of mode C_i(x) of the vorticity ω and instantaneous wavenumber k(x) is separated by the forcing scale k_f into two parts, which correspond to the inverse energy cascade and the forward enstrophy cascade. It is found that all conditional probability density function p(C|k) at given wavenumber k has an exponential tail. In the inverse energy cascade, the shape of p(C|k) does collapse with each other, indicating a nonintermittent cascade. The measured scaling exponent ζ _{ω }^I(q) is linear with the statistical order q, i.e., ζ _{ω }^I(q)=-q/3, confirming the nonintermittent cascade process. In the forward enstrophy cascade, the core part of p(C|k) is changing with wavenumber k, indicating an intermittent forward cascade. The measured scaling exponent ζ _{ω }^F(q) is nonlinear with q and can be described very well by a log-Poisson fitting: ζ _{ω }^F(q)=1/3q+0.45left( 1-0.43^qright). However, the extracted vorticity scaling exponents ζ_ω(q) for both inverse energy cascade and forward enstrophy cascade are not consistent with Kraichnan's theory prediction. New theory for the vorticity field in 2D turbulence is required to interpret the observed scaling behavior.