Turbulence Noise

We show that the large-eddy motions in turbulent uid ow obey a modiied hydrody-namic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity eld itself. We do so by means of an exact procedure of \statis-tical ltering" of the Navier-Stokes equations, which formally solves the closure problem, and we discuss relations of our analysis with the \decimation theory" of Kraichnan. We show that the statistical ltering procedure can be formulated using eld-theoretic path-integral methods within the Martin-Siggia-Rose formalism for classical statistical dynamics. We also establish within the MSR formalism a \least-eeective-action principle" for mean turbulent velocity prooles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in eld-theory|which characterizes the static eeective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of elds|is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity prooles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g. a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochas-tic subgrid modelling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.