Lagrangian chaos and the enstrophy cascade in Ekman-Navier-Stokes two-dimensional turbulence

TL;DR

This study investigates how Ekman friction alters the direct enstrophy cascade and the Lagrangian chaotic statistics in 2D turbulence. It combines high-resolution Ekman–Navier–Stokes simulations with long-time Lagrangian trajectory analysis to characterize Finite Time Lyapunov Exponent (FTLE) statistics and to derive a phenomenological model for the Lyapunov exponent $\lambda$ across friction strengths. The authors show that FTLE fluctuations are effectively Gaussian in the high-friction limit, derive a Kraichnan-like relation $\lambda = Z/(2\alpha)$, and use a quadratic Cramér-function approximation to predict the spectral slope correction $\xi$, agreeing well with numerical spectra. The work highlights the dominant role of forcing-scale dynamics under strong friction and provides a framework for predicting spectral corrections from Lagrangian statistics through a controlled interpolation between friction regimes.

Abstract

Two-dimensional turbulence with linear (Ekman) friction exhibits spectral properties that deviate from the classical Kraichnan prediction for the direct enstrophy cascade. In particular, for sufficiently small viscosity and large friction, the enstrophy flux is suppressed in the cascade and, as a consequence, the small-scale vorticity field becomes passively transported by the large-scale, chaotic flow. We numerically address this problem by investigating how the statistics of the Lagrangian Finite Time Lyapunov Exponent in 2D Ekman-Navier-Stokes simulations are affected by the friction coefficient and by the other parameters of the flow. We derive a simple phenomenological model that interpolates the dependence of the Lyapunov exponent on the flow statistics from the large friction limit, where analytical predictions are available, to the small friction region. We find that the distribution of the FTLE around this mean value is always close to a Gaussian, and this allows to make a simple prediction for the correction of the spectral slope of the direct cascade which is in very good agreement with the numerical results.

Figures (6)

• Figure 1: Enstrophy dissipation rate due to friction $\eta_{\alpha}$ (main) and viscosity $\eta_{\nu}$ (inset), both normalized by enstrophy input $\eta_{I}$ and shown as functions of nondimensionalised friction.
• Figure 2: (a) Lyapunov exponent as a function of nondimensional friction. The blue line represents the high-friction prediction $\lambda = \eta_I/4\alpha^2$. (b) Lyapunov exponent as a function of the square root of enstrophy (nondimensionalised). The red line shows Eq. \ref{['model_lyap']} with coefficients $A=6.780$, $B=-3.985$ and $C=0.597$.
• Figure 3: Cramér functions for all $\alpha\in[0.16, 2.56]$ computed at time $T\lambda \simeq 10$. Both axes are normalized to collapse the different curves according to the Gaussian approximation. The darkest points correspond to the lowest friction parameter $\alpha$, and the black line represents $x^2/2$. Note that the Gaussian distribution does not capture the tails for the smaller values of friction (darkest points), indicating a non-negligible skewness.
• Figure 4: Friction dependence of the Cramér functions' parameter $a$ and $b$.
• Figure 5: (a) Energy spectra $E(k)$ compensated with the dimensional scaling $k^3$ and (b) corresponding enstrophy flux $\Pi(k)$ for $\alpha=[0.16, 0.32, 0.64, 0.96, 1.28, 1.60, 2.00, 2.56]$ (from top to bottom). Simulations at resolution $N=8192$.