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 B_nF_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 ζ_h, characterizes the eddy diffusivity and can take values in the interval 0≤ζ_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 ζ₃ of the zero modes in the limit ζ_h→0, and we display quantitative agreement with the nonperturbative calculations in this limit. Another interesting limit is ζ_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 ζ₃ vanishes as √ζ₂/|Inζ₂, where ζ₂ is the scaling exponent of the second-order correlation function. In this limit as well, perturbative calculations are consistent with the nonperturbative calculations.