Introduction to Geophysical Flows Abstract: Atmospheric winds and oceanic currents are strongly influenced by rotation, density stratification, and their small aspect ratio. We will present emblematic examples highlighting peculiar properties of these flows, with an emphasis on phenomena taking place at the equator. A useful general reference for the lectures is the book "Atmospheric and Oceanic Fluid Dynamics", 2nd Edition 2017, Cambridge University Press, by Geoffrey Vallis. Lecture 1: Rotation We will focus on a striking consequence of rotation : the trapping of planetary waves along the equator. The discovery of this waveguide in 1966 by Matsuno was the culmination of two centuries of endeavour on the computation of dynamical tides. It is nowadays a central piece in our understanding of climate phenomena such as El Nino. Keywords: shallow water equations, Coriolis parameter, beta-plane approximation, Rossby waves, Inertial oscillations, Poincaré waves, Kelvin waves, Rossby number, Rossby radius, Laplace tidal equations. References : [1] Matsuno, Taroh. "Quasi-geostrophic motions in the equatorial area."Journal of the Meteorological Society of Japan. Ser. II 44, no. 1 (1966): 25-43. [2] Gallagher, Isabelle, and Laure Saint-Raymond. "On the influence of the Earth’s rotation on geophysical flows." Handbook of mathematical fluid dynamics 4 (2007): 201-329. Lecture 2: Curvature We will provide a semi-classical interpretation to the equatorial wave spectrum, relating the existence of two unidirectional modes computed by Matsuno to peculiar topological properties of a dual wave problem introduced by Kelvin in 1880. It turns out that equatorial dynamics bears strong similarities with topological insulators, which are exotic materials that have attracted a lot of attention from physicists over the last decades. Keywords: Berry curvature, Chern number, Bulk-interface correspondence, Wigner/Weyl transforms, Ray tracing. References: [1] Cheverry, Christophe, Isabelle Gallagher, Thierry Paul, and Laure Saint-Raymond. "Semiclassical and spectral analysis of oceanic waves." (2012): 845-892 [2] Faure, Frédéric. "Manifestation of the topological index formula in quantum waves and geophysical waves." arXiv preprint arXiv:1901.10592 (2019). [3] Delplace, Pierre, J. B. Marston, and Antoine Venaille. "Topological origin of equatorial waves." Science 358, no. 6366 (2017): 1075-1077. [4] Venaille, Antoine, Yohei Onuki, Nicolas Perez, and Armand Leclerc. "From ray tracing to waves of topological origin in continuous media." SciPost Physics 14, no. 4 (2023): 062.