Abstract: In this talk, we will present some recent mathematical results, mainly the global well-posedness and convergence of the relaxation limit, on two kinds of dynamical models for the atmosphere with moisture. In the first part of this talk, which is a joint work with Edriss S. Titi , we will consider a tropical atmosphere model introduced by Frierson, Majda, and Pauluis (Commum. Math. Sci. 2004); for this model, we will present the global well-posedness of strong solutions and the strong convergence of the relaxation limit, as the relaxation time $/varepsilon$ tends to zero. It will be shown that, for both the finite-time and instantaneous-relaxation systems, the $H^1$ regularities on the initial data are sufficient for both the global existence and uniqueness of strong solutions, but slightly more regularities than $H^1$ are required for both the continuous dependence and strong convergence of the relaxation limit. In the second part of this talk, which is a joint work with Sabine Hittmeir, Rupert Klein, and Edriss S. Titi , we will consider a moisture model for warm clouds used by Klein and Majda (Theor. Comput. Fluid Dyn. 2006), where the phase changes are allowed, and we will present the global well-posedness of this system.
 Jinkai Li; Edriss S. Titi: A tropical atmosphere model with moisture: global well-posedness and relaxation limit, Nonlinearity, 29 (2016), no. 9, 2674--2714.
 Sabine Hittmeir; Rupert Klein; Jinkai Li; Edriss S. Titi: Global well-posedness for passively transported nonlinear moisture dynamics with phase changes, 30 (2017), no. 10, 3676--3718.