Anomalous scaling of anisotropy of second-order moments in a model of a randomly advected solenoidal vector field

A model of randomly advected solenoidal field is presented. The model is formally derived by a linearization of the Navier-Stokes equation with respect to the perturbation to a basic state and by assuming the characteristic time scale of the basic state to be very short. The model includes a nonlocal (in space) effect through a pressurelike term that keeps the advected field solenoidal, but still yields exact equations for multipoint moments. The advecting field is assumed to be statistically homogeneous and isotropic with zero mean and structure function with exponent xi. An analysis is made of the scaling of the steady second-order moments of the solenoidal field in two dimensions. The scaling exponent zeta(l) of the isotropic part (l=0) and the anisotropic part for the angular wave number l=2 is obtained analytically or numerically. The scaling of the isotropic part does not depend on whether the pressurelike term is present or not while the scaling of the anisotropic part is affected by the pressurelike term. There are two homogeneous similarity solutions with real positive exponents zeta(2) when xi>xi(c)(2) approximately 1.3. The same kind of analysis is also applied to a simplified two-point closure equation