Derivation of a Generalized Quasi-Geostrophic Approximation for Inviscid Flows in a Channel Domain: The Fast Waves Correction

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 Bourgeois and Beale (SIAM J Math Anal 25(4):1023–1068, 1994. https://doi.org/10.1137/S0036141092234980 )] or domain with no boundary [as in Embid and Majda (Commun Partial Differ Equ 21(3–4):619–658, 1996. https://doi.org/10.1080/03605309608821200 )]. 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.