Incompressible Fluid Turbulence & Generalized Solutions of Euler Equations

Gregory Eyink

Department of Applied Mathematics and Statistics
Whiting School of Engineering
The Johns Hopkins University
Department of Mathematics

Experiments and numerical simulations show that energy dissipation in incompressible fluid turbulence tends to a positive value in the inviscid limit (infinite Reynolds number). Lars Onsager (1949) proposed an explanation for this phenomenon in terms of energy cascade for certain singular solutions of Euler equations. We shall review current ideas on the nature of turbulent energy cascade and their status within rigorous theory of PDE's. In particular, we shall discuss a classical picture of Geoffrey Taylor (1937) on the role of vortex line-stretching in generating turbulent energy dissipation. Taylor's argument was based on a statistical hypothesis that material lines in a turbulent flow will tend to elongate, on average, and appealed to the Kelvin Theorem (1869) on conservation of circulations. For smooth solutions the Kelvin Theorem for all loops is equivalent to the Euler equations of motion, but we shall present rigorous results which suggest that the theorem breaks down in turbulent flow due to nonlinear effects. This turbulent "cascade of circulations" has been verified by high-Reynolds-number numerical simulations. We propose another conjecture, that circulations on material loops may be martingales of a generalized Euler flow (in the sense of Brenier and Shnirelman). We shall show that this property has a close analogue in the "Kraichnan model" of random advection, which accounts for anomalous scalar dissipation in that model. The "Kraichnan model" is also known to probabilists as a generalized stochastic flow and its basic features have been put on a rigorous footing by Le Jan and Raimond (2002, 2004). We propose a geometric treatment of this model, formally as a diffusion process on an infinite-dimensional semi-group of volume-preserving maps.