Persistence in Advection of Passive Scalar

Details Persistence in Advection of Passive Scalar Persistence in Advection of Passive Scalar

2008-10-02

arxiv.org/abs/0810.0512v1

Physics

Condensed Matter

Statistical Mechanics

4 pages

Scientific paper

10.1103/PhysRevE.79.031112

We consider the persistence phenomenon in advectecd passive scalar equation in 1-dimension. The velocity field is random with the $ \sim |k|^{-(2+\alpha)}$. In presence of the non-linearity the complete Green's function becomes $G^{-1}=-i\omega+Dk^2+\Sigma$. 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.

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