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I shall discuss two interesting PDE problems for the tropical atmosphere. Using systematic multiscale asymptotics, we arrive at an asymptotic closure for the ideal fluid equations governing dynamics on large scales in the tropical atmosphere. Using a plausible model for smaller scale flows in the tropics, we are able to calculate the structure of the Madden-Julian oscillation; this is a planetary scale organization of winds,the understanding of which has been called "the holy grail" of tropical meteorology. In the second problem, we study the same equations, but over longer time and spatial scales. The resultant coupled nonlinear dispersive equations for the amplitudes of interacting wave packets are novel both from the perspective of the atmospheric sciences and from a more general mathematical setting. These equations describe the influence of large scale tropical waves on midlatitude waves and, in particular, are relevant for understanding the effect of the Madden-Julian oscillation on midlatitude weather. Furthermore, the amplitude equations have a Hamiltonian structure and admit analytic solitary wave solutions.