Universal direct cascade in two-dimensional turbulence.

Abstract

We consider the correlation functions of vorticity \ensuremath{\omega} in the region of the direct cascade in a steady two-dimensional turbulence. The nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates, and the main problem is to take those logarithms into account. Our procedure starts directly with the Euler equation rewritten in the comoving reference frame. We express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain. It enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval. We find the indices characterizing the logarithmic behavior of different correlation functions. For example, the two-point simultaneous functions are as follows: 〈${\mathrm{\ensuremath{\omega}}}^{\mathit{n}}$ (${\mathbf{r}}_{1}$)${\mathrm{\ensuremath{\omega}}}^{\mathit{n}}$(${\mathbf{r}}_{2}$)〉\ensuremath{\sim}[${\mathit{P}}_{2}$ln(L/${\mathbf{r}}_{1}$-${\mathbf{r}}_{2}$\ensuremath{\Vert})${]}^{2\mathit{n}3}$, where L is the pumping scale. We demonstrate that the form of those correlation functions is universal, i.e., independent of the pumping. The only pumping-related value which enters the expressions is the enstrophy production rate ${\mathit{P}}_{2}$. The contributions related to pumping rates ${\mathit{P}}_{\mathit{n}}$ of the higher-order integrals of motion are demonstrated to be small in comparison with the ones induced by ${\mathit{P}}_{2}$. We establish also the time dependence of the correlation functions, the correlation time \ensuremath{\tau} in the comoving reference frame is the same for the vorticity and strain and is scale dependent: \ensuremath{\tau}\ensuremath{\propto}${\mathrm{ln}}^{2/3}$(L/r). We reformulate our procedure in the diagrammatic language to reinforce the conclusions.