Diffusion of a passive scalar in two-dimensional turbulence

Abstract

We study the spectral statistics of the fluctuations of a passive scalar convected by a two-dimensional homogeneous isotropic turbulence using the eddy-damped quasinormal Markovian (EDQNM) theory, and – in certain cases – the near-equivalent test-field model (TFM). For zero correlation between scalar and vorticity fields it is known that these closures lead to inertial–convective ranges following respectively a k−1 law in the enstrophy-cascade range and a k−5/3 law in the inverse-energy-cascade range. We show that the scalar cascade in the latter range is direct, and is characterized by a positive eddy diffusivity. For forced flows in which correlation between vorticity and scalar forcing is prescribed, the k−5/3 range is replaced by a k1/3 range if the correlation is perfect, and for imperfect correlation we describe an analysis that bridges the ranges k1/3, k−1.We also examine the infrared (k → 0) behaviour of the energy and scalar spectra. Statistically steady injection of energy and scalar variance at a wavenumber kI produces an energy spectrum E(k) ∼ k3, and a scalar spectrum E0(k) ∼ k. In the unforced case, for any initial conditions, the energy spectrum develops a k3 range for k less than the wavenumber characteristic of the energy-containing eddies and a k−3 range for larger k. This allows a demonstration – via closure – that this energy-bearing wavenumber decreases as t−1 and the enstrophy as t−2 (modulo logarithmic corrections), as predicted by Batchelor (1969). Finally we show the scalar-fluctuation variance decays as the enstrophy, if the enstrophy spectrum is considered as a passive scalar. If not, the decay exponent is proportional to the ratio of the characteristic eddy-damping rates of the velocity and scalar third-order moments.