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

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.