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On the problem of beam focusing in the turbulent atmosphere

Two parts of the problem were analyzed. The first one is the adequate description of turbulence. The result is the simulation of the evolution of the refractive index due to turbulence. The second one is the beam focusing on condition that the refractive index is subject to spatial and temporal variations. The turbulence was simulated with the aid of the solution to the Navier-Stokes equations. Three kinds of initial conditions were used: (i) the vortical field was given, the velocity divergence (dilatation) being zero and the temperature being constant; (ii) the velocity divergence was given, the vorticity being zero and the temperature being constant; (iii) there is a temperature distribution, the vorticity and the dilatation being zero. In all cases, the initial values of the density are constant. The problem is set in the infinite space, the initial data being random functions. The solution of the Navier- Stokes equations was reduced to the solution of integral equations of the Volterra type. The iterative procedure was used. The comparison of the subsequent iterations allows to conclude that the convergence takes place. The problem of compensation for turbulent distortions of a laser beam was solved. The resolving function determines the necessary deformation of the mirror. The knowledge of the resolving function indicates the way to the beam focusing in the turbulent atmosphere.