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Abstract
The subject of incompressible fluid flows in rapidly rotating cavities, relevant to the dynamics of the Earth's outer core, is addressed here by means of numerical modeling. We recall in the introduction what makes this topic fascinating and challenging, and emphasize the need for new, more flexible numerical approaches in line with the evolution of today's parallel computers. Relying upon recent advances in numerical analysis, we first introduce in chapter 2 a spectral element model of the axisymmetric Navier-Stokes equation, in a rotating reference frame. Comparisons with analytical or published numerical solutions are made for various test problems, which highlight the spectral convergence properties and adaptivity of the approach. In chapter 3, we couple this axisymmetric kernel with a Fourier expansion in longitude in order to describe the dynamics of three-dimensional convection flows. Again, several reference problems are studied. In the specific case of a rotating fluid undergoing thermal convection, this so-called Fourier-spectral element method (FSEM) proves to be as accurate as standard pseudo-spectral techniques. Having this numerical tool anchored on solid grounds, we study in chapter 4 fluid flows driven by thermal convection and precession at the same time. A new topic in the vast field of fluid mechanics, convecto-precessing flows are of particular importance for the Earth's core, and the equations governing their evolution are derived in detail. We solve these using the FSEM; results seem to indicate that to first order, thermal convection and precession ignore each other. We discuss the relevance of these calculations for the Earth's core and outline directions for future related research.
