Mathematics for Fluid Modeling

The chair’s research focuses on the mathematical modeling of fluids, and more specifically on geophysical fluids, which are characterized by the involvement of multiple time and spatial scales and by high anisotropy. This anisotropy—which in this context refers to an asymmetry between the horizontal and vertical directions—has three main sources:

• The geometry of ocean basins, whose aspect ratio (depth to horizontal extent) is extremely small;
• The Earth’s rotation, which imposes strong vertical stiffness on the fluid;
• Vertical stratification in density, salinity, and temperature.

It gives rise to a vast array of dynamic and thermodynamic phenomena, both in terms of mean currents and in terms of perturbations (waves and long-period oscillations). Added to this are boundary effects (coasts, ocean floors, and the free surface), including ocean-atmosphere couplings.

The objective of the chair’s research is to address this complexity by isolating the dominant phenomena at each scale and by introducing elementary mathematical models for each of these phenomena. The analysis of partial differential equations—their well-posedness, stability, qualitative properties, and asymptotic analysis under specific parameter regimes—is central to this research. However, the research also draws upon a wide variety of mathematical fields, including spectral theory, uncertainty quantification, scientific computing, control theory, dynamical systems, and the study of bifurcations…

The chair’s teaching activities aim to synthesize recent advances in these topics at the interface between mathematics, physics, and Earth sciences, and to produce reference materials that enable researchers and students in these various disciplines to acquire this body of knowledge. In the coming years, these teaching activities will be organized around three main areas:

• Dynamics (waves, walls and boundary layers, interfaces)
• Thermodynamics (compressibility effects, congestion effects, couplings with dynamics, complex fluids)
• Instabilities and tipping points