Multiscale Dynamics of 2D Rotational Compressible Euler Equations — an Analytical Approach

We study the 2D rotational compressible Euler equations with two singular parameters: the Rossby number for rotational forcing and the Froude/Mach number for pressure forcing. The competition of these two forces leads to a newly found parameter δ = τσ−2 that serves as a characteristic scale separating two dynamic regimes: δ 1 for the strong rotation regime [CT08] and δ 1 for the mild/weak rotation regime. The analytical novelty of this study is correspondingly two-fold. In the δ 1 regime, we utilize the method of iterative approximations that starts with the pressureless rotational Euler equations previously studied in [LT04]. The resulting approximation is a periodic-in-time flow that reflects the domination of rotation in this small regime. On the other hand, for δ 1, we combine fast wave analysis for nonlinear hyperbolic PDEs with Strichartz-type estimates to reveal an approximate incompressible flow. Our argument is highlighted with newly established nonlinear invariants in terms of wave interaction and is free of Fourier analysis.