Well-posedness and long-term behaviour for a troposphere wave propagation model

TL;DR

This paper analyzes a modified troposphere wave propagation model, establishing its well-posedness and long-term behavior, including conditions for the existence of attractors or runaway solutions, by leveraging its relation to the 2D primitive equations.

Contribution

It extends the understanding of a recent troposphere wave model by proving well-posedness and characterizing long-term dynamics, including attractor existence and parameter-dependent solutions.

Findings

Global well-posedness for certain parameters

Existence of a global attractor under specific conditions

Runaway solutions with unbounded growth for other parameters

Abstract

In this paper, we investigate a model recently derived by A. Constantin and R.S. Johnson for nonlinear wave propagation in the troposphere, particularly the 'morning glory' cloud pattern. We consider the model with natural Dirichlet boundary conditions for the vertical velocity at the top of the troposphere, and thus introduce a new pressure term. This modified system has a structural relation to the 2D primitive equations, for which global well-posedness and the existence of a global attractor are already known. We transfer these results to the modified model, giving proofs that exploit specific features and use standard methods combined with anisotropic Sobolev inequalities. Additionally, we show that the attractor exists only for specific parameter ranges, while for other parameters, we find runaway solutions with unbounded growth over time.