Anomalous Scaling Exponents of a Passive Scalar Advected by Turbulence

Abstract

For Kraichnan’s problem of passive scalar advection by a velocity field delta-correlated in time, any simultaneous correlation function of a scalar satisfies a closed differential equation so that all common hypotheses about intermittency could, in principle, be verified by direct calculation. In an isotropic turbulence, the n-point correlation function depends on n(n − l)/2 distances, which makes direct solution of the respective partial differential equation quite difficult if the parameters are not at all constrained. In the limit of large space dimensionality d ≫ 1, the anomalous exponents are small and are found by perturbation theory in 1/d. We demonstrate how the anomalous part of the many-point correlation funstion appears as a zero mode of the operator of turbulent diffusion which exploits the the interchange symmetry between the points. We then consider passive scalar convected by multi-scale turbulent velocity field with short yet finite temporal correlations. Taking the limit of a white velocity as a zero approximation we develop perturbation theory with respect to a small velocity correlation time τ. We show that the anomalous scaling exponents of the scalar field continuously depend on τ i.e. they are nonuniversal.