Multiscale dynamics of 2D rotational compressible Euler equations

We study the 2D rotational incompressible Euler equations with two singular parameters: the Rossby number τ for rotational forcing and the Froude/Mach number σ for pressure forcing. The competition of these two forces leads to a newly found parameter δ = τσ−2 that serves as a characateristic scale separating two dymamic regimes: δ 1 for the strong rotation regime ([1]) and δ 1 for the mild/weak rotation regime ([2]). 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 [3]. The resulting approximation is a periodic-in-time flow that reflects the domination of rotation in this small δ 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 an invariant-based analysis on the nonlinear interations of fast waves and therefore is potentially easy to generalize. Reference