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

TL;DR

This paper investigates the scale at which observational data can effectively reconstruct small-scale flows in two-dimensional Navier-Stokes turbulence, revealing that this scale is closer to the forcing scale rather than the dissipation scale, unlike in three dimensions.

Contribution

The study introduces the concept of an essential resolution length scale in 2D turbulence and compares it with the 3D case, highlighting differences due to turbulence dynamics.

Findings

In 2D turbulence, the essential resolution scale is near the forcing scale.

The scale for successful reconstruction in 2D is much larger than the dissipation scale.

Differences between 2D and 3D scales are explained by inter-scale interactions and cascades.

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.