Characterizing Data Assimilation in Navier-Stokes Turbulence with Transverse Lyapunov Exponents

TL;DR

This paper introduces a theoretical framework using transverse Lyapunov exponents to characterize data assimilation in turbulent flows, identifying a critical length scale for successful synchronization and linking it to turbulence properties.

Contribution

It proposes a novel chaos-based approach to analyze data assimilation in turbulence, connecting Lyapunov exponents with the effectiveness of reconstructing small-scale structures.

Findings

Identifies a critical length scale for synchronization in turbulence.

Links the critical length scale to the Reynolds number.

Suggests new directions for data assimilation algorithms.

Abstract

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.