O. Gat et al., PERTURBATIVE AND NONPERTURBATIVE ANALYSIS OF THE 3RD-ORDER ZERO MODESIN THE KRAICHNAN MODEL FOR TURBULENT ADVECTION, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 56(1), 1997, pp. 406-416
The anomalous scaling behavior of the nth-order correlation functions
F-n of the Kraichnan model of turbulent passive scalar advection is be
lieved to be dominated by the homogeneous solutions (zero modes) of th
e Kraichnan equation beta(n)F(n)=0. In this paper we present an extens
ive analysis of the simplest (nontrivial) case of n=3 in the isotropic
sector. The main parameter of the model, denoted as zeta(h), characte
rizes 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 appro
priate variables we can present nonperturbative numerical calculations
of the zero modes in a projective two dimensional circle. In this pre
sentation 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 cal
culations in this limit. Another interesting limit is zeta(h)-->2. Thi
s 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-or
der correlation function. In this limit as well, perturbative calculat
ions are consistent with the nonperturbative calculations.