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

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 analysis, in the second part of the thesis, we examine the effects of high rotation (small Rossby number) for the 2-D incompressible density-dependent Euler system. With respect to the previous problem, now we deal with an incompressible system with a hyperbolic structure, where the viscosity effects are neglected. More precisely, the main goal is to perform the singular limit in the fast rotation regime, showing the convergence of the Euler equations to a quasi-homogeneous type system. The limit system is a coupled system of a transport equation for the density and a momentum equation for the velocity with a non-linear term of lower order, which combines the effects of fluctuations of the density and the velocity field. For the convergence process, a core point is to develop uniform (with respect to Ro) estimates in high regularity norms not to deteriorate the lifespan of solutions. Moreover, as a sub-product of the local well-posedness analysis (recall that the global existence of solutions is an open problem even in 2-D), we find an "asymptotically global" well-posedness result: for small densities, the lifespan of solutions to the primitive and limiting systems tend to infinity. The proof of convergence of the two primitive problems (the Navier-Stokes-Fourier system and the Euler system, respectively) towards the reduced models is based on a compensated compactness argument. The key point is to use the structure of the underlying system of Poincaré waves in order to identify some compactness properties for suitable quantities. Compared to previous results, our method enables to treat the whole range of parameters in the multi-scale problem, and also to reach and go beyond the "critical" choice Fr=\sqrt{Ma}.

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 analysis, in the second part of the thesis, we examine the effects of high rotation (small Rossby number) for the 2-D incompressible density-dependent Euler system. With respect to the previous problem, now we deal with an incompressible system with a hyperbolic structure, where the viscosity effects are neglected. More precisely, the main goal is to perform the singular limit in the fast rotation regime, showing the convergence of the Euler equations to a quasi-homogeneous type system. The limit system is a coupled system of a transport equation for the density and a momentum equation for the velocity with a non-linear term of lower order, which combines the effects of fluctuations of the density and the velocity field. For the convergence process, a core point is to develop uniform (with respect to Ro) estimates in high regularity norms not to deteriorate the lifespan of solutions. Moreover, as a sub-product of the local well-posedness analysis (recall that the global existence of solutions is an open problem even in 2-D), we find an "asymptotically global" well-posedness result: for small densities, the lifespan of solutions to the primitive and limiting systems tend to infinity. The proof of convergence of the two primitive problems (the Navier-Stokes-Fourier system and the Euler system, respectively) towards the reduced models is based on a compensated compactness argument. The key point is to use the structure of the underlying system of Poincaré waves in order to identify some compactness properties for suitable quantities. Compared to previous results, our method enables to treat the whole range of parameters in the multi-scale problem, and also to reach and go beyond the "critical" choice Fr=\sqrt{Ma}.