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

Abstract In this paper, we investigate a model recently derived by A. Constantin and R.S. Johnson for non-linear wave propagation in the troposphere, particularly the ‘morning glory’ cloud pattern. We consider the model with free-slip boundary conditions at the top and bottom boundaries, 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, in the absence of forcing, we find runaway solutions with unbounded growth over time.