Perturbative and Non-Perturbative Analysis of the 3'rd Order Zero Modes

The anomalous scaling behavior of the n-th 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 $\hat{B}_n F_n=0$. In this paper we present an extensive analysis of the simplest (non-trivial) 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 $0\le \zeta_h \le 2$. After choosing appropriate variables we can present computer-assisted non-perturbative 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 $\zeta_h\to 0$, and we display quantitative agreement with the non-perturbative calculations in this limit. Another interesting limit is $\zeta_h\to 2$. 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 like $\sqrt{\zeta_2/log{\zeta_2}}$. In this limit as well, perturbative calculations are consistent with the non-perturbative calculations.