Stability, well-posedness and blow-up criterion for the Incompressible Slice Model

TL;DR

This paper proves local existence, uniqueness, and blow-up criteria for the Incompressible Slice Model, a modified atmospheric model based on Hamilton's principle, with implications for weather prediction and atmospheric front formation.

Contribution

It introduces a new well-posedness and stability analysis for the ISM, extending the Cotter-Holm Slice Model to the Euler-Boussinesq case.

Findings

Proved local existence and uniqueness of strong solutions.

Established a blow-up criterion for the ISM.

Analyzed Arnold's stability around certain equilibria.

Abstract

In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter-Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.