Stability of scaling regimes in $d>~2$ developed turbulence with weak anisotropy

Fully developed turbulence with weak anisotropy is investigated by means of the renormalization group approach and double expansion regularization for dimensions $d>~2.$ Some modification of the standard minimal subtraction scheme has been used to analyze the stability of the Kolmogorov scaling regime which is governed by the renormalization group fixed point. This fixed point is unstable at $d=2;$ thus the infinitesimally weak anisotropy destroys the above scaling regime in two-dimensional space. The restoration of the stability of this fixed point, under a transition from $d=2\mathrm{}$ to 3, is demonstrated at a borderline dimension $2<\mathrm{}{d}_c<3.\mathrm{}$ The results are in qualitative agreement with results recently obtained in the framework of a typical analytical regularization scheme.