Tutorial

SO(3) Analysis

Luca Biferale
Dip. di Fisica
Universita' di Tor Vergata

The problem of anisotropy and its effects on the statistical theory of high Reynolds-number (Re) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last five years.

In this talk we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations.

Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO($d$) symmetry group (with $d$ being the dimension, $d=2$ or 3). This device allows a discussion of the scaling properties of the statistical objects in well defined sectors of the symmetry group, each of which is determined by the "angular momenta" sector numbers $(j,m)$. For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made:

1. the scaling exponents are universal,
2. the isotropic scaling exponents are always leading,
3. the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of $j$.

This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales.

Next we explain how to apply the SO(3) decomposition to the statistical Navier-Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite.

A systematic analysis of Direct Numerical Simulations (DNS) of the Navier-Stokes equations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of $j$. The conflicting experimental measurements on the decay of anisotropy are explained and systematized, showing agreement with the theory presented here.