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

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