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In the present thesis, we are interested in the description of the dynamicsof flows on large scales. In this context, the fluids are governed byrotational, weak compressibility and stratification effects, whose importanceis measured by adimensional numbers: the Rossby, Mach and Froude numbers. Thefirst part of the thesis is dedicated to the analysis of a 3D multi-scaleproblem called the full Navier-Stokes-Fourier system where variations indensity and temperature are considered and in addition we take into account theCoriolis, centrifugal and gravitational forces. We study, in the framework ofweak solutions, the combined incompressible and fast rotation limits in theregime of small Mach, Froude and Rossby numbers and for general ill-preparedinitial data. In the case when the Mach number is of higher order than theRossby number, we prove that the limit dynamics is described by anincompressible Oberbeck-Boussinesq type system. Conversely, when the Mach andRossby numbers have the same order of magnitude, we show convergence towards aquasi-geostrophic type equation. Following "le fil rouge" of the asymptoticanalysis, in the second part of the thesis, we examine the effects of highrotation for the 2D incompressible density-dependent Euler system. Now, themain goal is to perform the singular limit in the fast rotation regime, showingthe convergence of the Euler equations to a quasi-homogeneous type system. Theproof of convergence of the two primitive problems towards the reduced modelsis based on a compensated compactness argument. The key point is to use thestructure of the underlying system of Poincar\'e waves in order to identifysome compactness properties for suitable quantities. Compared to previousresults, our method enables to treat the whole range of parameters in themulti-scale problem, and also to reach and go beyond the critical choice$Fr=\sqrt{Ma}$.

Some stability and instability issues in the dynamics of highly rotating fluids