Synchronisation in two-dimensional damped-driven Navier-Stokes turbulence: insights from data assimilation and Lyapunov analysis

TL;DR

This study investigates how partial observations can synchronise two-dimensional Navier–Stokes turbulence by reconstructing unobserved small scales from large-scale data. Using continuous data assimilation and a conditional Lyapunov exponent framework, it identifies a critical observation wavenumber $k_a^*$ that marks successful synchronization. For Kolmogorov forcing with Ekman drag, the results show $k_a^{* (2D)}$ is of order the forcing scale $k_f$, in stark contrast to 3D where $k_a^{* (3D)}\approx 0.2/\eta$ near the dissipation scale, and the authors attribute this to nonlocal interscale interactions and orbital instabilities. The findings imply that, in 2D, small-scale fields can be reconstructed from relatively coarse observations, guiding experimental and numerical strategies for quasi-two-dimensional turbulent systems and informing future extensions to rotating/stratified flows.

Abstract

In Navier--Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional NS equations indicate that employing observational data resolved down to a specific length scale, $\ell^{3D}_{\ast}$, enables the successful reconstruction of small-scale flows. Such a length scale of `essential resolution of observation' for reconstruction $\ell^{3D}_{\ast}$ is close to the dissipation scale in three-dimensional NS turbulence. % Here we study the equivalent length scale in {\it two}-dimensional NS turbulence, $\ell^{2D}_{\ast}$, and compare with the three-dimensional case. Our numerical studies using data assimilation and conditional Lyapunov exponents reveal that, for Kolmogorov flows with Ekman drag, the length scale $\ell^{2D}_{\ast}$ is actually close to the forcing scale, substantially larger than the dissipation scale. Furthermore, we discuss the origin of the significant relative difference between the length scales, $\ell^{2D}_{\ast}$ and $\ell^{3D}_{\ast}$, based on inter-scale interactions, `cascades' and orbital instabilities in turbulence dynamics.

Figures (8)

• Figure 1: Schematic of orbital instability and data assimilation in a chaotic dynamical system. Orbital instability expands uncertainty (dashed ellipses), whereas data assimilation contracts uncertainty (solid ellipses) along the trajectory. The figure illustrates a case in which data assimilation succeeds, with the uncertainty contracting exponentially along the orbit, characterised by the negative conditional Lyapunov exponent $\lambda_c~(<0)$, as $e^{\lambda_c t}$.
• Figure 2: Data assimilation experiment for two-dimensional Navier--Stokes turbulence. Snapshots of vorticity fields at times $t = 0, 1, 2$, and $3$, from top to bottom: the reference DNS field $\bm{u}(t) = \bm{p}(t) + \bm{q}(t)$; the observational data $\bm{p}(t)$ obtained by low-pass filtering the DNS field with $k_a = 4$; and the field obtained through data assimilation (CDA) $\tilde{\bm{u}}(t) = \bm{p}(t) + \tilde{\bm{q}}(t)$. A corresponding video is available as supplementary material.
• Figure 3: Time series of enstrophy $\Omega(t)$ (top), with a close-up view for $0 \le t \le 10$ (inset), and the enstrophy-norm error $\Delta\Omega(t)$ as defined in (\ref{['eq:errordef']}) (bottom). The dashed lines show the convergence rates given by the conditional Lyapunov exponents $\lambda_c(k_a)$.
• Figure 4: Same as figure \ref{['enstrophy']}, with the vertical axis showing the palinstrophy, $P(t)$, and the palinstrophy-norm error, $\Delta P(t)$.
• Figure 5: Synchronisation process examined over scales. The solid lines show the time evolution of the energy spectrum of the difference of the fields $\Delta E(k,\,t)$ obtained through the CDA process for the case $k_a = 4$. From thickest to thinnest, the curves correspond to times $t = 0,\, 10,\, 50,\, 100,\, 150,\, \text{and}\ 200$. The dashed line shows the result of the DNS, corresponding to the long-time-averaged energy spectrum $\langle E(k) \rangle_T$. The inset shows the scaled energy spectra, $\Delta E(k,\,t)e^{2 \lambda_c t}$, plotted with the same colours.