Anomalous scaling of a passive vector field in $d$ dimensions: Higher-order structure functions

Abstract

The problem of anomalous scaling in the model of a transverse vector field $\theta_{i}(t,x)$ passively advected by the non-Gaussian, correlated in time turbulent velocity field governed by the Navier--Stokes equation, is studied by means of the field-theoretic renormalization group and operator product expansion. The anomalous exponents of the $2n$-th order structure function $S_{2n}(r) = <[\theta(t,x) - \theta (t,x+ r)]^{2n}>$, where $\theta$ is the component of the vector field parallel to the separation $r$, are determined by the critical dimensions of the family of composite fields (operators) of the form $(\partial\theta\partial\theta)^{2n}$, which mix heavily in renormalization. The daunting task of the calculation of the matrices of their critical dimensions (whose eigenvalues determine the anomalous exponents) simplifies drastically in the limit of high spatial dimension, $d\to\infty$. This allowed us to find the leading and correction anomalous exponents for the structure functions up to the order $S_{56}$. They reveal intriguing regularities, which suggest for the anomalous exponents simple "empiric" formulae that become practically exact for $n$ large enough. Along with the explicit results for modest $n$, they provide the full description of the anomalous scaling in the model. Key words: passive vector field, turbulent advection, anomalous scaling, renormalization group, operator product expansion.