Scaling exponents in weakly anisotropic turbulence from the Navier–Stokes equation

The second-order velocity structure tensor of weakly anisotropic strong turbulence is decomposed into its SO(3) invariant amplitudes d j ( r ). Their scaling is derived within a scaling approximation of a variable-scale mean-field theory of the Navier–Stokes equation. In the isotropic sector j = 0 Kolmogorov scaling d 0 ( r ) ∝ r 2/3 is recovered. The scaling of the higher j amplitudes ( j even) depends on the type of the external forcing that maintains the turbulent flow. We consider two options: (i) for an analytic forcing and for decreasing energy input into the sectors with increasing j , the scaling of the higher sectors j > 0 can become as steep as d j ( r ) ∝ r j +2/3 ; (ii) for a non-analytic forcing we obtain d j ( r ) ∝ r 4/3 for all non-zero and even j .