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 $<v(k,\omega )v(-k,-\omega )>\sim |k{|}^{-(2+\alpha )}$. In presence of the non-linearity the complete Green's function becomes $G^{-1}=-i\omega +D{k}^2+\Sigma$. We determine $\Sigma$ self-consistently from the correlation function which gives $\Sigma \sim k^{\beta }$, with $\beta =(1-\alpha )/2$. The effect of the non-linear term in the equation in the $\mathcal{O}({\epsilon }^2)$ is to replace the diffusion term due to molecular viscosity by an effective term of the form ${\Sigma }_0{k}^{\beta }$. The stationary correlator for this system is $[\mathrm{Sech}(T/2){]}^{1/\beta }$. Using the self-consistent theory we have determined the relation between $\beta$ and $\alpha$. Finally, IIA is used to determine the persistent exponent.

4 pages

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Cited by in corpus (5)

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