The direct cascade of two-dimensional hydrodynamic turbulence in a square box with periodic boundary conditions along both coordinates in the frame of the solution of equation for vorticity are numerically investigated in the presence of pumping and viscous-like damping. Equation for vorticity was solved numerically using pseudospectral Fourier method, while integration in time was performed with the use of a hybrid Runge--Kutta/Crank?Nicholson third-order scheme. The initial conditions were chosen as random sets of Gaussian shape vortices with zero mean vorticity and randomly distributed over the entire domain. Simulations were performed at the Computer Center of the Novosibirsk State University (with the use of the NVIDIA CUDA technology). The spatial resolution was up to 16384x16384.
Numerical results are shown that the formation of a power-law dependence on wave number k in the Kraichnan-type spectrum of turbulence formed owing quasi-shocks of vorticity is a very quick process. If at an early stage (at the time of order of the inverse of the pumping increment), the development of turbulence is about the same scenario as in the case of a freely decaying turbulence. Formed quasi-singular distribution of di-vorticity, which in k-space correspond to jets - Fourier transforms of quasi-shocks, leading to a strong turbulence anisotropy. In the next much slower stage, the structure of quasi-shocks lines is complicated. The distances between quasi-shocks lines are reduced, and the spectrum becomes more isotropic. It is also important to note that the probability distribution function of vorticity at these times there is a formation of exponential tail at large arguments, which can be extrapolated as a linear dependence of vorticity in accordance with the theoretical predictions. The probability distribution function for di-vorticity also has two specific areas: the first - the distribution function is close to Poisson PDF, the second (large value of di-vorticity) - distribution function is exponential behavior with more distinct linear dependence than similar for vorticity. Both of these observations suggest that direct cascade of turbulence at large times loses anisotropy due to a tendency to breaking. In our opinion, there are at least two possible reasons of turbulence isotropization. The first reason may be related to the pumping area, where, in spite of the strong dissipation at low k, formed large-scale vortices (it's some remains not killed until the end of inverse cascade), which, due to its rotation, contribute into the system of vorticity quasi-shocks additional stretching of di-vorticity lines, and on the other hand - makes the system of significant lines of the di-vorticity field more complicated. We observed that the isotropic spectrum is carried out by time, more than time of Kraichnan-type enstrophy transferring, when pumping wave reaches the viscous region. As is known, the direct cascade of turbulence is a non-local (or rather - weakly nonlocal), that is accompanied by the appearance of logarithmic corrections to the Kraichnan spectrum. Locality of turbulence means that the main nonlinear interaction is interaction between scale of the same order. Interaction greatly different scales is strongly depressed. In this situation, both boundary to the inertial interval areas - pumping and viscous dissipation, as isotropic sources, in our opinion, are responsible for the turbulence isotropization of direct cascade.