Passive Scalar: Scaling Exponents and Realizability

Abstract

An isotropic passive scalar field $T$ advected by a rapidly-varying velocity field is studied. The tail of the probability distribution $P(θ,r)$ for the difference $θ$ in $T$ across an inertial-range distance $r$ is found to be Gaussian. Scaling exponents of moments of $θ$ increase as $\sqrt{n}$ or faster at large order $n$, if a mean dissipation conditioned on $θ$ is a nondecreasing function of $|θ|$. The $P(θ,r)$ computed numerically under the so-called linear ansatz is found to be realizable. Some classes of gentle modifications of the linear ansatz are not realizable.