Noise in Fluids

Fluids, in complex regimes, show random features. The aim of this project is approaching several questions around the randomness of fluids by means of a theory that could be called “Stochastic Fluid Mechanics”. The distinctive feature of this theory, opposite to others that investigated the stochastic features of fluids, is that it is based on the usual continuum mechanics equations, in particular the Navier-Stokes and Euler equations, but suitably modified by the presence of random elements, like an additive or a transport type noise.Stochastic equations of fluid dynamics have been studied already for three decades and the number of foundational results is very large. However, two basic directions have been explored only partially:a) the origin and the form of noise in fluidsb) the consequences of the presence of noise.This project will make progresses in these two directions, describing the noise near boundary due to vortex productions, including the question of intrinsic stochasticity at the boundary, the propagation of additive noise at small scales to a transport-stretching noise at large scales, the consequences of transport noise on eddy viscosity, enhanced dissipation, enhanced coalescence, and other applications in turbulence and Geophysics. The most ambitious core of the project is putting together these pieces in a picture that explains the mechanism of regularization by noise for the 3D Navier-Stokes equations. The additive noise at small scales is responsible for a transport-stretching noise at larger scales which could prevent blow-up of high intensity vortex structures. We have already proved recently that a noise, of transport type only, has this regularization effect, but stretching amplifies vorticity and new progresses are needed to cope with both processes. We aim to use the experimentally observed fact that small scale velocity should be approximately orthogonal to vorticity in high intensity regions.