  This paper is devoted to investigating the rotating Boussinesq equations of
inviscid, incompressible flows with both fast Rossby waves and fast internal
gravity waves. The main objective is to establish a rigorous derivation and
justification of a new generalized quasi-geostrophic approximation in a channel
domain with no normal flow at the upper and lower solid boundaries, taking into
account the resonance terms due to the fast and slow waves interactions. Under
these circumstances, We are able to obtain uniform estimates and compactness
without the requirement of either well-prepared initial data (as in [10]) or
domain with no boundary (as in [17]). In particular, the nonlinear resonances
and the new limit system, which takes into account the fast waves correction to
the slow waves dynamics, are also identified without introducing Fourier series
expansion. The key ingredient includes the introduction of (full) generalized
potential vorticity.
