Irreversibility of the two-dimensional enstrophy cascade

We study the time irreversibility of the direct cascade in two-dimensional turbulence by looking at the time derivative of the square vorticity along Lagrangian trajectories, a quantity called metenstrophy. By means of extensive direct numerical simulations we measure the time irreversibility from the asymmetry of the probability density function of the metenstrophy and we find that it increases with the Reynolds number of the cascade, similarly to what is found in three-dimensional turbulence. A detailed analysis of the different contributions to the enstrophy budget reveals a remarkable difference with respect to what is observed for the energy cascade, in particular the role of the statistics of the forcing to determine the degree of irreversibility.

Figure 1

Probability density functions of metenstrophy $q$ at increasing $\text{Re}$ (from the inner to the outer PDF).

Figure 2

Second (red circles) and third (blue squares) moments of the PDFs of metenstrophy $q$ as a function of $\text{Re}$. The inset shows the skewness of metenstrophy as a function of $\text{Re}$.

Figure 3

The PDF of the metenstrophy $q$ for the case $\text{Re}=1.5\times {10}^4$ (back solid line) and of the different contributions: viscosity $q_{\nu }$ (red dashed line), friction $q_{\alpha }$ (green dotted line), and forcing $q_f$ (blue dash-dotted line).

Figure 4

Snapshots of the vorticity field $\omega$ (top) and viscous enstrophy dissipation rate $q_{\nu }$ (bottom) at the same time for the simulation at $\text{Re}=3.7\times {10}^3$.

Figure 5

The PDFs of metenstrophy $q$ for different correlation times ${\tau }_f$(${\tau }_f/{\tau }_{\omega }$ increases from the outer to the inner curve).

Figure 6

Second (red circles) and third (blue squares) moments of the PDF of metenstrophy $q$ as a function of the forcing correlation time ${\tau }_f$. The inset shows the skewness of metenstrophy as a function of ${\tau }_f$.

Figure 7

The PDF of the metenstrophy $q$ for the case ${\tau }_f{\eta }^{1/3}=0.15$ (back solid line) and of the different contributions: viscosity $q_{\nu }$ (red dashed line), friction $q_{\alpha }$ (green dotted line), and forcing $q_f$ (blue dash-dotted line).