Extended stability timescales for the 2D3C rotating stratified Boussinesq equations

We study the time evolution of a vertical-slice model for incompressible inhomogeneous rotating fluids perturbed around a stably stratified rest state with linear buoyancy profile. For ε-small localized initial data with vanishing potential vorticity, we prove that solutions remain of size O(ε) up to times of order ε⁻⁴³, and up to times of order ε⁻² in a regime where the Brunt--Väisälä frequency and the Coriolis parameter are comparable. The first result relies on linear dispersion with amplitude decay rate t⁻¹², while the second exploits the absence of three-wave resonances via a normal form transformation. Our results remain valid for small nonzero initial potential vorticity, whose size is propagated by means of an intrinsic change of coordinates.