Isotropization of two-dimensional hydrodynamic turbulence in the direct cascade

We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to). If at the earlier stage (at the time of order of the inverse pumping growth rate $\tau\sim\Gamma_{\max}^{-1}$), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence $[1, 2]$: quasi-singular distribution of di-vorticity are formed, which in $k$-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of $10\tau$ turbulence becomes almost isotropic. In particular, at these times any significant anisotropy in the angular fluctuations for the energy spectrum (for a fixed $k$) is not visible, while the probability distribution function of vorticity for large arguments has the exponential tail with the exponent linearly dependent on vorticity, in the agreement with the theoretical prediction [3].