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

The anomalous scaling behavior of the $n$th-order correlation functions ${\mathcal{F}}_n$ of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation $\mathcal{B}{⁁}_n{\mathcal{F}}_n=0\mathrm{}$. In this paper we present an extensive analysis of the simplest (nontrivial) case of $n=3\mathrm{}$ 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 $0<~{\zeta }_h<~2$. 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 ${\zeta }_h\to 0$, and we display quantitative agreement with the nonperturbative 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 as $\sqrt{{\zeta }_2/|\ln {\zeta }_2|}$, 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.