Some properties of two-dimensional inverse energy cascade dynamics

In this work we analyze the degree of homogeneity and stationarity of the transfers in the inverse energy cascade of two-dimensional turbulence. Two extreme cases, namely, a well-developed inverse energy cascade in a robust statistically steady state and the collision of two vortices of the same sign, which is a clear illustration of a nonstationary cascade regime, are studied. We consider the absolute transfer ${\mathrm{\eta }}_{\mathrm{l}}$ at scale l produced by the nonlinear term of the Navier-Stokes equation. The scaling properties of the transfer hierarchy 〈η${\mathrm{}}_{\mathrm{l}}^{\mathrm{p}+1\mathrm{}}$〉/〈η${\mathrm{}}_{\mathrm{l}}^{\mathrm{p}}$〉∼${\mathrm{l}}^{\mathrm{-}{\mathrm{\delta }}_{\mathrm{p}}}$ are examined. We define Δ=(${\mathrm{\delta }}_{\mathrm{\infty }}$-${\mathrm{\delta }}_0$)/${\mathrm{\zeta }}_3^{\mathrm{*}}$, where ${\mathrm{\zeta }}_3^{\mathrm{*}}$ is the scaling of the third-order structure function of absolute velocity increments, ${\mathrm{\delta }}_0$ is a quantity tracing the smallest but most frequent transfers, and ${\mathrm{\delta }}_{\mathrm{\infty }}$ characterizes the largest but rarest transfers. We show that Δ plays a fundamental role in the scaling description of the cascade dynamics. In two-dimensional energy cascade, the important property of the relationship between the scaling of the structure functions and the distribution of the heterogeneities in the physical space given by (${\mathrm{\delta }}_{\mathrm{\infty }}$ -${\mathrm{\delta }}_0$ ) is the invariance of Δ. Finally, we determine the physical meaning of the formally introduced adjustable parameters in She-Leveque [Phys. Rev. Lett. 72, 336 (1994)] and Dubrulle [Phys Rev. Lett. 73, 7 (1994); 73, 959 (1994)] intermittency models.