David Ruelle, IHES

Abstract: The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system. Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents tau_p and zeta_p associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note, is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments. Specifically, if p(z)dz is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number increases, log p(z) passes from a concave to a linear then to a convex profile for large z as observed. We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents tau_p and zeta_p.