Advection and the Efficiency of Spectral Energy Transfer in Two-Dimensional Turbulence

We report measurements of the geometric alignment of the small-scale turbulent stress and the large-scale rate of strain that together lead to the net flux of energy from small scales to large scales in two-dimensional turbulence. We find that the instantaneous alignment between these two tensors is weak and, thus, that the spectral transport of energy is inefficient. We show, however, that the strain rate is much better aligned with the stress at times in the past, suggesting that the differential advection of the two is responsible for the inefficient spectral transfer. We provide evidence for this conjecture by measuring the alignment statistics conditioned on weakly changing stress history. Our results give new insight into the relationship between scale-to-scale energy transfer, geometric alignment, and advection in turbulent flows.

Figure 1

(a) Mean spectral energy flux ${\mathrm{\Pi }}^{(r)}$ as a function of the filter scale $r$ scaled by the magnet spacing $L_m$. Negative values correspond to a flux of energy to larger length scales. (b) Mean instantaneous angle ${\mathrm{\Theta }}_{s\tau }^{(r)}$ (in units of $\pi$) between the eigenframes of the stress ${\tau }_{ij}^{(r)}$ and the strain rate $s_{ij}^{(r)}$ as a function of filter scale $r$. The dashed line shows 45°.

Figure 2

Average of ${\mathrm{\Theta }}_{s\tau }^{(r)}(\mathrm{\Delta }t)$, the angle between the eigenframes of the stress at one time and the strain rate at a time $\mathrm{\Delta }t$ later, for several different filter scales $r$. Each curve shows ${\mathrm{\Theta }}_{s\tau }^{(r)}(\mathrm{\Delta }t)$ for a different $r$, and the colors correspond to the colors of the symbols in Fig. 1. The dashed line shows 45°.

Figure 3

Probability density functions (PDFs) of ${\mathrm{\Theta }}_{s\tau }^{(r)}(\mathrm{\Delta }t)$ at $r=2L_m$ (the most intense inverse energy flux) for positive and negative $\mathrm{\Delta }t$. For increasing positive $\mathrm{\Delta }t$, the peak of the PDF shifts toward more alignment; however, for increasing negative $\mathrm{\Delta }t$, the peak shifts toward misalignment, and the PDF ultimately becomes uniform.

Figure 4

Conditional averages of ${\mathrm{\Theta }}_{s\tau }^{(r)}$ (see text for the definition) as a function of $\alpha$, the multiplier of the ensemble-averaged stress-angle standard deviation. Smaller values of $\alpha$ correspond to less prior variation in the orientation of the stress eigenframe. Data are shown for three filter scales: $r=L_m$ (squares), $r=2L_m$ (circles), and $r=3L_m$ (diamonds). Error bars show the standard error of the mean. In all cases, weaker prior variability of the stress eigenframe orientation corresponds to stronger instantaneous alignment between the stress and the rate of strain.