Non-equivalence of quasilinear dynamical systems and their statistical closures

TL;DR

This paper challenges the assumption that statistical closures like CE2 accurately replicate the statistics of quasilinear dynamical systems, revealing fundamental differences due to rank instabilities.

Contribution

It provides counterexamples showing that CE2, although an exact closure for QL dynamics, can produce statistics that differ from the actual QL solutions due to rank instabilities.

Findings

CE2 predictions can disagree with QL simulation statistics.

Rank instabilities in CE2 cause discrepancies.

Counterexamples in 2D fluid systems demonstrate non-equivalence.

Abstract

It is widely believed that statistical closure theories for dynamical systems provide statistics equivalent to those of the governing dynamical equations from which the former are derived. Here, we demonstrate counterexamples in the context of the widely used mean-field quasilinear (QL) approximation applied to 2D fluid dynamical systems. We compare statistics of QL numerical simulations with those obtained by direct statistical simulation via a cumulant expansion closed at second order (CE2). We observe that, though CE2 is an exact statistical closure for QL dynamics, its predictions disagree with the statistics of the QL solution for identical parameter values. These disagreements are attributed to instabilities, which we term rank instabilities, of the second cumulant dynamics within CE2 that are unavailable in the QL equations.