Aula Volterra, Scuola Normale Superiore
In this talk I will discuss how the framework of geometric continuum mechanics can be made stochastic by introducing the stochastic Euler-Poincare theorem. This naturally leads to noise of transport type in fluid dynamics while maintaining geometric invariants, such as the enstrophy in two-dimensional ideal fluids. In special cases, it is possible to discretise the equations of motion while preserving most of the structure, which provides strong numerical evidence for Kraichnan’s double cascade conjecture in two-dimensional incompressible fluids. I will also discuss some recent progress for the quasi-geostrophic equations.