Persistence in Advection of Passive Scalar

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 $G^{-1}=-i\omega+Dk^2+\Sigma$. 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.