  Data assimilation (DA) reconstructing small-scale turbulent structures is
crucial for forecasting and understanding turbulence. This study proposes a
theoretical framework for DA based on ideas from chaos synchronization, in
particular, the transverse Lyapunov exponents (TLEs). The analysis with TLEs
characterizes a critical length scale, below which the turbulent dynamics is
synchronized to the larger-scale turbulent dynamics, indicating successful DA.
An underlying link between TLEs and the maximal Lyapunov exponent suggests that
the critical length scale depends on the Reynolds number. Furthermore, we
discuss new directions of DA algorithms based on the proposed framework.
