The semi-geostrophic (SG) equations are essential for modeling the evolution of large-scale wind fronts. Optimal transport theory offers a compact interpretation of these equations and some notion of weak solution. In the first part of this talk, I will present the connections that arise between optimal transport and fluid dynamics, following Yann Brenier’s interpretation of both Arnold’s work on the Euler equations from the 1960s and the formulation of the SG equations by Hoskins, Cullen, and their coauthors in the 1970s-1980s.

Building on this foundation, I will then discuss how the entropic approximation method provides a practical numerical resolution of the SG equations through the Sinkhorn algorithm. This approach corresponds to a PDE approximation of the SG system. I will present well-posedness results for this PDE and analyze the convergence of the entropic scheme as both the regularization parameter and the discretization step vanish.

This presentation is based on joint works with J.-D. Benamou and C. Cotter (JCP, 2023) as well as G. Carlier (arxiv, 2024)