The Slow Invariant Manifold of the Lorenz--Krishnamurthy Model

TL;DR

This paper applies the Flow Curvature Method to analytically derive high-order approximations of the slow invariant manifold in the Lorenz--Krishnamurthy model, surpassing previous low-order results and confirming invariance.

Contribution

It introduces a novel application of the Flow Curvature Method to obtain up to eighteenth-order approximations of the slow manifold in the Lorenz--Krishnamurthy model, extending prior work.

Findings

Eighteenth-order approximation of the slow manifold achieved.

Thirteenth-order approximation for the conservative model obtained.

Invariance of the manifolds established via Darboux theorem.

Abstract

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic procedures involved for such a construction diverge. So, it seems that till now only the first-order and third-order approximations of this slow manifold have been analytically obtained. While using the Flow Curvature Method we show in this work that one can provide the eighteenth-order approximation of the slow manifold of the generalized Lorenz-Krishnamurthy model and the thirteenth-order approximation of the "conservative" Lorenz-Krishnamurthy model. The invariance of each slow manifold is then established according to Darboux invariance theorem.