arXiv:nlin/0101050v1 [nlin.CD] 29 Jan 2001 Stability of scaling regimes in d ≥ 2 developed turbulence with weak anisotropy M. Hnatich 1 , E. Jonyova 1,2 , M. Jurcisin 1,2 , M. Stehlik 1 1 Permanent address: Institute of Experimental Physics, SAS, Koˇ sice, Slovakia , 2 Joint Institute for Nuclear Research, Dubna, Russia Abstract The fully developed turbulence with weak anisotropy is investigated by means of renormalization group approach (RG) and double expansion regu- larization for dimensions d ≥ 2. Some modification of the standard minimal substraction scheme has been used to analyze stability of the Kolmogorov scal- ing regime which is governed by the renormalization group fixed point. This fixed point is unstable at d = 2; thus, the infinitesimally weak anisotropy de- stroyes above scaling regime in two-dimensional space. The restoration of the stability of this fixed point, under transition from d = 2 to d =3, has been demonstrated at borderline dimension 2 <d c < 3. The results are in qual- itative agreement with ones obtained recently in the framework of the usual analytical regularization scheme. 1 Introduction A traditional approach to the description of fully developed turbulence is based on the stochastic Navier-Stokes equation [1]. The complexity of this equation leads to great difficulties which do not allow one to solve it even in the simplest case when one assumes the isotropy of the system under consideration. On the other hand, the isotropic turbulence is almost delusion and if exists is still rather rare. Therefore, if one wants to model more or less realistic developed turbulence, one is pushed to consider anisotropically forced turbulence rather than isotropic one. This, of course, rapidly increases complexity of the corresponding differential equation which itself has to involve the part responsible for description of the anisotropy. An exact solution of the stochastic Navier-Stokes equation does not exist and one is forced to find out some convenient methods to touch the problem at least step by step. A suitable and also powerful tool in the theory of developed turbulence is the so-called renormalization group (RG) method 1 . Over the last two decades the RG 1 Here we consider the quantum-field renormalization group approach [2] rather than the Wilson renormalization group technique [3] 1