Chaotic Lagrangian models for turbulent relative dispersion

A deterministic multiscale dynamical system is introduced and discussed as a prototype model for relative dispersion in stationary, homogeneous, and isotropic turbulence. Unlike stochastic diffusion models, here trajectory transport and mixing properties are entirely controlled by Lagrangian chaos. The anomalous “sweeping effect,” a known drawback common to kinematic simulations, is removed through the use of quasi-Lagrangian coordinates. Lagrangian dispersion statistics of the model are accurately analyzed by computing the finite-scale Lyapunov exponent (FSLE), which is the optimal measure of the scaling properties of dispersion. FSLE scaling exponents provide a severe test to decide whether model simulations are in agreement with theoretical expectations and/or observation. The results of our numerical experiments cover a wide range of “Reynolds numbers” and show that chaotic deterministic flows can be very efficient, and numerically low-cost, models of turbulent trajectories in stationary, homogeneous, and isotropic conditions. The mathematics of the model is relatively simple, and, in a geophysical context, potential applications may regard small-scale parametrization issues in general circulation models, mixed layer, and/or boundary layer turbulence models as well as Lagrangian predictability studies.

Figure 1

The three regimes of the normalized FSLE $\lambda \left(\delta \right)/{\lambda }_L$, for an ideal 3D fully developed turbulent flow: (a) exponential separation, $\lambda \left(\delta \right)={\lambda }_L$, for $\delta \ll \eta$, where $\eta$ is the Kolmogorov length; (b) Richardson-Obukhov law, $\lambda \left(\delta \right)\sim {\delta }^{-2/3}$, inside the inertial range $\eta <\delta <L_0$, where $L_0$ is the integral length; (c) standard eddy diffusion, $\lambda \left(\delta \right)\sim {\delta }^{-2}$, at large separation scales $\delta \gg L_0$.

Figure 2

Collapse of the rescaled mean square trajectory separations computed in the QL configuration for different values of the Kolmogorov's constant $C_K$. Parameters $\epsilon$, $L_0$, and $T_0$ are (expressed in nondimensional units) turbulent dissipation rate, maximum eddy size, and integral time scale, respectively (a). Empirical scaling relation between Kolmogorov constant ($C_K$) and Richardson's constant ($C_R$) of the model (b). Richardson's constant $C_R$ is computed by fitting the Richardson's law, at given $C_K$ and $\epsilon$, to the mean square relative separation $\langle r^2\left(t\right)\rangle$ inside the inertial range.

Figure 3

Normalized FSLE $\lambda \left(\delta \right)/{\lambda }_L$, with fixed rate of energy dissipation $\epsilon$, fixed Kolmogorov length $\eta$ and four different values of the integral length scale $L_0$: (a) $L_0/\eta \sim 10$ (diamonds); (b) $L_0/\eta \sim {10}^2$ (triangles); (c) $L_0/\eta \sim {10}^3$ (circles); and (d) $L_0/\eta \sim {10}^4$ (squares). Statistics over $N=8000$ particle pairs with uniform random initial positions. Left panel: E model; right panel: QL model.

Figure 4

FSLE for an eight-decade inertial range, $L_0/\eta \sim {10}^8$: QL model (squares) and E model (circles). Statistical errors are of the same order as the size of the symbols. The QL model always satisfies the Richardson-Obukhov law $\sim {\delta }^{-2/3}$ inside the inertial range while the E model displays an anomalous $\sim {\delta }^{-1/3}$.

Figure 5

Upper panel: $p_R(r,t)$ at three different times $t=t_1,t_2,t_3$ measured in Kolmogorov time unit (symbols) as function of $r$ normalized to the standard deviation; the continuous line is the Richardson pdf $p_R^{*}(r,t)$; the dotted line is a Gaussian pdf. Lower panel: Collapse of rescaled $p_e\left(\tau \right)$ at three different separations $\delta =r_1,r_2,r_3$ within the inertial range (symbols); the line fitting the exponential tail is $p_e^{*}\left(\tau \right)$.