Anomalous scaling for passively advected magnetic fields.

Abstract

The Kraichnan model [1] for passive scalar advection is by now considered a paradigm for intermittency problems. A first systematic understanding of the scalar field scaling properties has been attained. The crucial property of the model is that equal-time correlation functions obey closed equations of motion. This stems from the small correlation time of the velocity field υ advecting the scalar. It has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions [2, 3, 4]. Such mechanism can be illustrated in its simplest form by considering the following linear differential equation $$\frac{{{d^2}y(r)}}{{d{r^2}}} + \frac{a}{r} \frac{{dy(r)}}{{dr}} + b\frac{{y(r)}}{{{r^2}}} = f(\frac{r}{L})$$ (1) where r varies on the positive axis, y(r) is the unknown function, a and b are two constants and f (r/L) varies over distances of order L and is almost constant for r ≪ L. In the physical situations, L is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale. As for intermittency in Navier-Stokes turbulence, we are interested in the scaling behavior in the inertial range, i.e. for r ≪ L. Note that the l.h.s. of (1) is invariant under scale transformations, reflecting the scale-invariance of υ. The general solution of (1) is given by a non-homogeneous and a linear combination of two homogeneous solutions. The combination is fixed by the conditions that y(r), being a correlation function, should be regular and vanish at infinity.