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.
Categories: physics.flu-dyn math.DS nlin.CD