Analytic calculation of the anomalous exponents in turbulence: Using the fusion rules to flush out a small parameter

The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorov’s statistics of 1941 (K41) acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents ${\zeta }_n$ of the $n\mathrm{th}$-order structure functions can be computed analytically. The crux of the method consists of renormalizing a four-point interaction amplitude on the basis of the fusion rules. This amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of ${\zeta }_2,$ ${\delta }_2={\zeta }_2-2/3\approx \mathrm{0.03.}$ Higher-order interaction amplitudes are shown to be even smaller. The corrections to K41 to $O\left({\delta }_2\right)$ result from standard logarithmically divergent ladder diagrams in which the four-point interaction acts as a “rung.” The theory allows a calculation of the anomalous exponents ${\zeta }_n$ in powers of the small parameter ${\delta }_2.$ The n dependence of the scaling exponents ${\zeta }_n$ stems from pure combinatorics of the ladder diagrams. In this paper we calculate the exponents ${\zeta }_n$ up to $O\left({\delta }_2^3\right).\mathrm{}$ Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter ${\delta }_2.$ The actual evaluation of the small parameter ${\delta }_2$ from first principles requires additional developments that are outside the scope of this paper.