Characterizing data assimilation in Navier-Stokes turbulence with transverse Lyapunov exponents

Data assimilation of turbulent systems is crucial for many applications, including weather prediction, and related to the turbulence physics and mathematical properties of the Navier-Stokes equations, such as the concept of the inertial manifold. We propose a mathematical framework to study the assimilation phenomena characterized by the transversal stability of an invariant manifold containing a turbulent attractor, which we call a data-assimilation manifold. The transversal stability of the manifold is quantified by the transverse Lyapunov exponents. The Lyapunov analysis of the three-dimensional Navier-Stokes turbulence clarifies that the orbital instability of the turbulent attractor determines the occurrence of the data assimilation phenomena, particularly the critical length scale for the slaving small-scale dynamics.