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 $\omega(x,y)$ 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 $8192^2$ 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 $\omega$ and instantaneous wavenumber $k(x)$ is separated by the forcing scale $k_f$ into two parts, which corresponding to the inverse energy cascade and the forward enstrophy cascade. It is found that all conditional pdf $p(C\vert k)$ at given wavenumber $k$ has an exponential tail. In the inverse energy cascade, the shape of $p(C\vert k)$ does collapse with each other, indicating a nonintermittent cascade. The measured scaling exponent $\zeta_{\omega}^I(q)$ is linear with the statistical order $q$, i.e., $\zeta_{\omega}^I(q)=-q/3$, confirming the nonintermittent cascade process. In the forward enstrophy cascade, the core part of $p(C\vert k)$ is changing with wavenumber $k$, indicating an intermittent forward cascade. The measured scaling exponent $\zeta_{\omega}^F(q)$ is nonlinear with $q$ and can be described very well by a log-Poisson fitting: $\zeta_{\omega}^F(q)=\frac{1}{3}q+0.45\left(1-0.43^{q}\right)$. However, the extracted vorticity scaling exponents $\zeta_{\omega}(q)$ for both inverse energy cascade and forward enstrophy cascade are not consistent with Kraichnan\rq{}s theory prediction. New theory for the vorticity field in 2D turbulence is required to interpret the observed scaling behavior.