Determining Modes, State Reconstruction, and Intertwinement: A Synchronization Framework

TL;DR

This paper introduces a new conceptual framework called self-synchronous intertwinement to clarify the relationship between determining modes and data assimilation filters in the 2D Navier-Stokes equations, supported by theoretical and numerical evidence.

Contribution

It develops a novel framework linking determining modes with synchronization in data assimilation, strengthening understanding of their interrelation in 2D NSE.

Findings

Self-synchronous intertwinement implies convergence of CDA filters.

Convergence of filters indicates finitely many determining modes.

Numerical experiments confirm theoretical results.

Abstract

This article studies the interrelation between the determining modes property in the two-dimensional (2D) Navier-Stokes equations (NSE) of incompressible fluids and the reconstruction property of two filtering algorithms for continuous data assimilation applied to the 2D NSE. These two properties are realized as manifestations of a more general phenomenon of "self-synchronous intertwinement." It is shown that this concept is a logically stronger form of asymptotic enslavement, as characterized by the existence of finitely many determining modes in the 2D NSE. In particular, this stronger form is shown to imply convergence of the direct-replacement filter and the nudging filter from continuous data assimilation (CDA), and then subsequently invoked to show that convergence in these filters implies that the 2D NSE possesses finitely many determining modes. The main achievement of this article is to therefore to develop a new conceptual framework, that of self-synchronous intertwinement, through which the precise inter-relationship between the determining modes property and synchronization phenomenon in these CDA filters is rigorously established and made decisively clear. The theoretical results are then complemented by numerical experiments that confirm the conclusions of the theorems.