Dimensional dependence of synchronisation in turbulence: insights from data assimilation and Lyapunov analysis

In Navier--Stokes (NS) turbulence, large-scale turbulent flows determine small-scale flows. Previous studies using data assimilation with the 3D NS equations indicate that employing observational data resolved down to a specific length scale L_3d enables the successful reconstruction of small-scale flows. Such a length scale of `essential resolution of observation' for reconstruction L_3d is close to the dissipation scale in 3D NS turbulence. Here we study the length scale in two-dimensional NS turbulence, L_2d, and compare with the 3D case.

The numerical studies using data assimilation and conditional Lyapunov exponents reveal that, for Kolmogorov flows with Ekman drag, the length scale L_2d is close to the forcing scale, which is substantially larger than the dissipation scale. Furthermore, we discuss the origin of the difference between the length scales, L_2d and L_3d, based on inter-scale interactions and orbital instabilities in turbulence dynamics.