Exact relation between Eulerian and Lagrangian velocity increment statistics

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Figure 1

(Color online) The plot shows $p_a\left(r\mid v_e;\tau \right)$ (top) and $p_a\left(v_p\mid v_e;\tau \right)$ (bottom) for $\tau =0.09T_I$.

Figure 2

The figure shows the impact of the transition probabilities on the Eulerian PDF. The left part of the figure shows $f_e\left(v_e;r\right)$ for $r=0.06,0.12,0.3$. These PDFs are transformed into $f\left(v_e;\tau \right)$ (middle) by the transition PDF presented in the upper half of Fig. 1. Subsequently $f\left(v_e;\tau \right)$ is converted into $f_l\left(v_l;\tau \right)$ (right picture) by the second transition PDF from Fig. 1. In both cases $\tau =0.09T_I$. In plots II and III the points denote the reconstructed PDFs based on Eqs. (9, 10), and the lines denote the PDFs directly computed from the Lagrangian data.

Figure 3

The figure shows $p_a\left(r;\tau \right)$ for $\tau =3.5{\tau }_{\eta },14{\tau }_{\eta },28{\tau }_{\eta }$ for three-dimensional turbulence. In the inset $p_a$ is depicted for the two-dimensional case with $\tau =0.22T_I,0.44T_I,0.9T_I$.