Renormalization of viscosity in wavelet-based model of turbulence

Statistical theory of turbulence in viscid incompressible fluid, described by the Navier-Stokes equation driven by random force, is reformulated in terms of scale-dependent fields u(a)(x), defined as wavelet-coefficients of the velocity field u taken at point x with the resolution a. Applying quantum field theory approach of stochastic hydrodynamics to the generating functional of random fields u(a)(x), we have shown the velocity field correlators < u(a1)(x(1)) ... u(an)(x(n))> to be finite by construction for the random stirring force acting at prescribed large scale L. The study is performed in d = 3 dimension. Since there are no divergences, regularization is not required, and the renormalization group invariance becomes merely a symmetry that relates velocity fluctuations of different scales in terms of the Kolmogorov-Richardson picture of turbulence development. The integration over the scale arguments is performed from the external scale L down to the observation scale A, which lies in Kolmogorov range 1 << A << L. Our oversimplified model is full dissipative: interaction between scales is provided only locally by the gradient vertex (u del)u, neglecting any effects or parity violation that might be responsible for energy backscatter. The corrections to viscosity and the pair velocity correlator are calculated in one-loop approximation. This gives the dependence of turbulent viscosity on observation scale and describes the scale dependence of the velocity field correlations.