  In this paper, we consider a nonlinear interaction system between the
barotropic mode and the first baroclinic mode of the tropical atmosphere with
moisture; that was derived in [Frierson, D.M.W.; Majda, A.J.; Pauluis, O.M.:
Dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation
limit, Commum. Math. Sci., 2 (2004), 591-626.] We establish the global
existence and uniqueness of strong solutions to this system, with initial data
in $H^1$, for each fixed convective adjustment relaxation time parameter
$\varepsilon>0$. Moreover, if the initial data enjoy slightly more regularity
than $H^1$, then the unique strong solution depends continuously on the initial
data. Furthermore, by establishing several appropriate
$\varepsilon$-independent estimates, we prove that the system converges to a
limiting system, as the relaxation time parameter $\varepsilon$ tends to zero,
with convergence rate of the order $O(\sqrt\varepsilon)$. Moreover, the
limiting system has a unique global strong solution, for any initial data in
$H^1$, and such unique strong solution depends continuously on the initial data
if the the initial data posses slightly more regularity than $H^1$. Notably,
this solves the VISCOUS VERSION of an open problem proposed in the above
mentioned paper of Frierson, Majda and Pauluis.
