Correlations between the instantaneous velocity gradient and the evolution of scale-to-scale fluxes in two-dimensional flow

Using high-resolution particle tracking velocimetry and filter-space techniques, we study the links between the scale-to-scale transfer of energy and enstrophy and instantaneously rotational and straining regions as determined by the classic Okubo-Weiss parameter in a quasi-two-dimensional laboratory flow. Although the Okubo-Weiss parameter has shortcomings, we find that, when suitably conditioned, it is surprisingly a good predictor for the future evolution of the spectral fluxes. By studying Lagrangian correlation functions, we explain our findings by showing that both the spectral fluxes and the Okubo-Weiss parameter are independently correlated for long times along fluid-element trajectories and thus that any predictive capacity of the Okubo-Weiss parameter arises because it is coupled to fluid advection. Our results suggest potential strategies for forecasting in complex flows by looking for quantities with long Lagrangian correlation times.

Figure 1

Conditional mean values of the (a) energy and (b) enstrophy fluxes as a function of the filter scale $r$ normalized by the magnet spacing $L_m$. Red squares denote mean values conditioned on being in rotational regions, whereas blue triangles denote those conditioned on being in straining regions. Black circles show the unconditional means.

Figure 2

Conditional skewness (normalized third moments) of the (a) energy and (b) enstrophy fluxes as a function of the filter scale $r$ normalized by the magnet spacing $L_m$. Red squares denote rotational regions, whereas blue triangles denote straining regions. Black circles show the unconditional skewnesses.

Figure 3

Time evolution of the Lagrangian averages of the (a) energy and (b) enstrophy fluxes at a fixed filter scale of $r=1.5L_m$. Black circles denote unconditional averages; red squares are averages taken over particles that were initially (at $\tau =0$) in rotational regions, whereas blue triangles are for those that were initially in straining regions.

Figure 4

Time evolution of the Lagrangian averages of the (a) energy and (b) enstrophy fluxes at a fixed filter scale of $r=2L_m$. Black circles denote unconditional averages; red squares are averages taken over particles that were initially in rotational regions, whereas blue triangles are for those that were initially in straining regions.

Figure 5

Lagrangian autocorrelation coefficient for the Okubo-Weiss parameter, filtered at different filter scales. The dashed black line shows the autocorrelation coefficient for the full (unfiltered) Okubo-Weiss parameter. The horizontal axis is normalized by $T_L$, the integral time scale for the velocity field. The inset shows the integral time scales for the different filter lengths. The colors of the symbols in the inset correspond to the filter scales used for the curves in the main panel.

Figure 6

Lagrangian cross correlations between the filtered Okubo-Weiss parameter and the (a) energy and (b) enstrophy fluxes. The colors are the same as in Fig. 5.