Perturbative and nonperturbative analysis of the third-order zero modes in the Kraichnan model for turbulent advection

The anomalous scaling behavior of the nth-order correlation functions F-n of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation beta(n)F(n)=0. In this paper we present an extensive analysis of the simplest (nontrivial) case of n=3 in the isotropic sector. The main parameter of the model, denoted as zeta(h), characterizes the eddy diffusivity and can take values in the interval 0 less than or equal to zeta(h) less than or equal to 2. After choosing appropriate variables we can present nonperturbative numerical calculations of the zero modes in a projective two dimensional circle. In this presentation it is also very easy to perform perturbative calculations of the scaling exponent zeta(3) of the zero modes in the limit zeta(h)-->0, and we display quantitative agreement with the nonperturbative calculations in this limit. Another interesting limit is zeta(h)-->2. This second limit is singular, and calls for a study of a boundary layer using techniques of singular perturbation theory. Our analysis of this limit shows that the scaling exponent zeta(3) vanishes as root zeta(2)/\1n zeta(2)\, where zeta(2) is the scaling exponent of the second-order correlation function. In this limit as well, perturbative calculations are consistent with the nonperturbative calculations.