A tropical atmosphere model with moisture: global well-posedness and relaxation limit

In this paper, we consider a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture, which was derived in Frierson et al (2004 Commum. Math. Sci. 2 591-626). We establish the global existence and uniqueness of strong solutions to this system, with initial data in H1, for each fixed convective adjustment relaxation time parameter ϵ 1. Moreover, if the initial data possess slightly more regularity than H 1, then the unique strong solution depends continuously on the initial data. Furthermore, by establishing several appropriate ϵ-independent estimates, we prove that the system converges to a limiting system as the relaxation time parameter ϵ tends to zero, with a convergence rate of the order O(√ϵ). Moreover, the limiting system has a unique global strong solution for any initial data in H1 and such a unique strong solution depends continuously on the initial data if the initial data posses slightly more regularity than H1. Notably, this solves the viscous version of an open problem proposed in the above mentioned paper of Frierson, Majda and Pauluis.