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In the present thesis, we are interested in the description of the dynamics of flows on large scales, like the atmospheric and ocean currents on the Earth. In this context, the fluids are governed by rotational, weak compressibility and stratification effects, whose importance is "measured" by adimensional numbers: the Rossby, Mach and Froude numbers respectively. More those three physical parameters are small, more the relative effects are strong. The first part of the thesis is dedicated to the analysis of a 3-D multi-scale problem called the full Navier-Stokes-Fourier system where variations in density and temperature are taken into account and in addition the dynamics is influenced by the action of Coriolis, centrifugal and gravitational forces. We study, in the framework of weak solutions, the combined incompressible and fast rotation limits in the regime of small Mach, Froude and Rossby numbers (Ma, Fr, Ro respectively) and for general ill-prepared initial data. In the so-called multi-scale regime where some effect is predominant in the motion, precisely when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq type system. It is worth noticing that the velocity field is purely horizontal at the limit (according to the so-renowned Taylor-Proudman theorem in geophysics), but surprisingly vertical effects on the temperature equation appear. These stratification effects are completely absent when Fr exceeds \sqrt{Ma}, whereas they suddenly come into play as soon as one reaches the endpoint scaling Fr=\sqrt{Ma}. Conversely, when the Mach and Rossby numbers have the same order of magnitude (the isotropic scaling), and in absence of the centrifugal force, we show convergence towards a quasi-geostrophic type equation for a stream-function of the limit velocity field, coupled with a transport-diffusion equation for a quantity that mixes the target density and temperature profiles. Following "le fil rouge" of the asymptotic ...
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