Abstract: "In Navier–Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional (3-D) NS equations indicate that employing observational data resolved down to a specific length scale, ℓ<sup>3-D</sup><sub>∗</sub>, enables the successful reconstruction of small-scale flows. Such a length scale of ‘essential resolution of observation’ for reconstruction ℓ<sup>3-D</sup><sub>∗</sub> is close to the dissipation scale in three-dimensional NS turbulence. Here, we study the equivalent length scale in two-dimensional (2-D) NS turbulence, ℓ<sup>2-D</sup><sub>∗</sub>, 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 ℓ<sup>2-D</sup><sub>∗</sub> 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, ℓ<sup>2-D</sup><sub>∗</sub> and ℓ<sup>3-D</sup><sub>∗</sub>, based on inter-scale interactions, ‘cascades’ and orbital instabilities in turbulence dynamics."
Original article: "Synchronisation in two-dimensional damped-driven Navier–Stokes turbulence: insights from data assimilation and Lyapunov analysis" - https://www.cambridge.org/core/journals/journal-of-fluid-mec...
Abstract: "In Navier–Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional (3-D) NS equations indicate that employing observational data resolved down to a specific length scale, ℓ<sup>3-D</sup><sub>∗</sub>, enables the successful reconstruction of small-scale flows. Such a length scale of ‘essential resolution of observation’ for reconstruction ℓ<sup>3-D</sup><sub>∗</sub> is close to the dissipation scale in three-dimensional NS turbulence. Here, we study the equivalent length scale in two-dimensional (2-D) NS turbulence, ℓ<sup>2-D</sup><sub>∗</sub>, 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 ℓ<sup>2-D</sup><sub>∗</sub> 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, ℓ<sup>2-D</sup><sub>∗</sub> and ℓ<sup>3-D</sup><sub>∗</sub>, based on inter-scale interactions, ‘cascades’ and orbital instabilities in turbulence dynamics."