Persistence in Advection of Passive Scalar

Abstract

We consider the persistence phenomenon in advectecd passive scalar equation in 1-dimension. The velocity field is random with the $ \sim |k|^{-(2+α)}$. In presence of the non-linearity the complete Green's function becomes $G^{-1}=-iω+Dk^2+Σ$. We determine $Σ$ self-consistently from the correlation function which gives $Σ\sim k^β$, with $β=(1-α)/2$. The effect of the non-linear term in the equation in the $\mathcal{O}(ε^2)$ is to replace the diffusion term due to molecular viscosity by an effective term of the form $Σ_0 k^β$. The stationary correlator for this system is $[\mathrm{Sech}(T/2)]^{1/β}$. Using the self-consistent theory we have determined the relation between $β$ and $α$. Finally, IIA is used to determine the persistent exponent.