Dissipation fluctuations of a passive scalar advected by a random velocity field.

Abstract

The dynamics of fluctuations of the dissipation rate of a passive scalar advected by a rapidly changing in time Gaussian random velocity field with the variance [${\mathit{v}}_{\mathit{i}}$(x)-${\mathit{v}}_{\mathit{i}}$(x+r)${]}^{2}$\ensuremath{\propto}${\mathit{r}}^{\ensuremath{\xi}}$ is considered. It is shown that when \ensuremath{\xi}/d\ensuremath{\rightarrow}0 the dissipation correlation function 〈E(x)E(x+r)〉\ensuremath{\propto}${\mathit{r}}^{\ensuremath{\gamma}}$ with \ensuremath{\gamma}=-4\ensuremath{\xi}/(d+2) in agreement with the recent works of Gawedzkii and Kupianen [Phys. Rev. Lett. 75, 3608 (1995)], and Chertkov et al. [Phys. Rev. E 52, 4924 (1995)]. It is shown that in this limit the fourth-order moment of the scalar difference is completely described in terms of the second-order moment and the scalar dissipation rate correlation function. \textcopyright{} 1996 The American Physical Society.