Statistics of Lagrangian trajectories in a rotating turbulent flow

We investigate the Lagrangian statistics of three-dimensional rotating turbulent flows through direct numerical simulations. We find that the emergence of coherent vortical structures because of the Coriolis force leads to a suppression of the “flight-crash” events reported by Xu et al. [Proc. Natl. Acad. Sci. (USA) 111, 7558 (2014)]. We perform systematic studies to trace the origins of this suppression in the emergent geometry of the flow and show why such a Lagrangian measure of irreversibility may fail in the presence of rotation.

Figure 1

Probability distribution functions of the vertical component of the vorticity ${\omega }_z$, normalized by its standard deviation ${\sigma }_z$ for $\text{Ro}=\infty$ (blue solid curve), 0.12 (red dashed curve), and 0.06 (black dashed-dotted curve)

Figure 2

Probability distribution functions of the fraction of time particles spend in vortical ($t_{Q_{+}}$) (filled markers) and straining ($t_{Q_{-}}$) (open markers) regions for $\text{Ro}=\infty$ (circles), 0.16 (stars), and 0.08 (triangles).

Figure 3

Probability distribution function of the energy increment $W\left(\tau \right)=E(t+\tau )-E\left(t\right)$, normalized by its root-mean-square value ${\langle W{\left(\tau \right)}^2\rangle }^{1/2}$ for $\tau /{\tau }_{\eta }=2$. The curves are for different values of Ro (see legend) and are artificially shifted by factors of 10 for clarity. Inset: The skewness of the pdfs of $W\left(\tau \right)$ as a function of Ro.

Figure 4

Representative plots of the symmetry function $S$ versus the energy increment $W\left(\tau \right)$ normalized by its root-mean-square value ${\langle W{\left(\tau \right)}^2\rangle }^{1/2}$ for $\tau /{\tau }_{\eta }=2$ for different values of Ro (see legend). Inset: Probability distribution functions of the negative (open symbols) and positive (filled symbols) values of the Lagrangian power for $\text{Ro}=\infty$, 0.08, and 0.06; the negative tail has been reflected for ease of comparison.

Figure 5

Probability distribution functions of the fraction of time during which tracers gain $t_{\mathrm{gain}}$ (filled circles) and lose $t_{\mathrm{loss}}$ (open circles) energy for (a) $\text{Ro}=\infty$, (b) $\text{Ro}=0.16$, and (c) $\text{Ro}=0.08$.

Figure 6

Representative plots, for $\text{Ro}=0.16$, of the pdf of the Lagrangian power, normalized by energy dissipation rate $\epsilon$, conditioned on whether the particles are in vortical $Q\ge 0$ (filled circles) or straining regions $Q<0$ (open circles). Inset: The skewness of the $Q$ showing a sharp increase with decreasing Rossby number.

Figure 7

The irreversibility Ir as a function of Ro showing a sharp increase, by an order of magnitude, as $\text{Ro}\to 1$.