Ludmila S. Shtemenko

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.

Modeling of laser beam propagation through turbulence

Our approach for modeling laser beam propagation through turbulence involves parabolic equation method and results of experimental investigation in laboratory. The analytic solution to the problem of the Gaussian beam propagation through non-uniform gas has been derived. The solution depends on the refracted index, i.e. on the gas density. The density distribution can be found from the Navier-Stokes system. The appropriate solution may be constructed by two ways : (i) as a series in powers of vorticity which is supposed to be small; (ii) with the aid of the parametrix method which includes an iterative procedure. It follows from the solution that acoustic radiation of vortex rings arises. Statistical properties of the propagating beam were found from the solution to the parabolic equation as average over time. In experiments the propagation path was equal to 7 m. The laser beam propagation was accompanied by convection and lateral wind. The frequency of turbulent fluctuations was equal to 2-10 Hz. Phase trajectories were found as well as statistical properties of the beam intensity in turbulent gas flow. The conclusion is as follows. Statistical characteristics traditionally used for the estimation of the laser beam special distortions in the open space transmission channels are to be complemented by the dynamic parameters such as the space of embeddings dimension, characteristic frequencies for the phase trajectories and so on.

Laser beam propagation through an ensemble of vortex rings in air

The problem of the evolution of an ensemble of vortex rings in air has been solved. The full system of the Navier -Stokes equations was used. The parametrix method was applied. The calculations were performed for a wide range of the ring parameters (circular and elliptic cross-sections, various diameters of the rings, their different orientation in the space etc.). The initial value problem is as follows. The vorticity has non-zero value only inside the rings at initial instant, the density and the temperature being constant everywhere at t=0. If the density is known, then it is possible to find the refractive index. The solution to the Navier-Stokes equations is an oscillating one. Thus the refractive index is an oscillating function with respect to time. These results enable to model turbulence in an adequate way without using the Taylor frozen turbulence hypothesis. The evolution of the frequency spectrum of the density fluctuations was obtained. These results were compared with Tatarskiis data. The intensity of a laser beam propagating through the ensemble of vortex rings in air was found with the aid of the parabolic equation method. A numerical procedure is set forth which allows to solve the problems of superresolution without using regularization methods. The task is as follows. There is a set of experimental data and an instrument function (with some error). We change the domain in such a manner that the corresponding MTF has nowhere zero values. The procedure enables to solve problems of focusing in the turbulent atmosphere.

Local-linear method of super-resolution for compensation of image distortions using new model of turbulence

Atmospheric turbulence is one of the important factors that influence on scene spatial resolution. In order to restore an image with minimum distortions one must know the correlation function for fluctuations of refractive index and the distorting PSF as a result. Grid-generated turbulence is a classic example of homogeneous and isotropic turbulence. Statistical properties of this flow have been investigated experimentally. In our case of grid-generated turbulence the statistical properties are distinct from the Kolmogorovs two-thirds law. Calculations performed on the basis of the linearized three-dimensional unsteady Navier-Stokes equations gave similar results. We modelled laser beam propagation through turbulent atmosphere and obtained numerical data for the distortion of images. The distortion of PSF and the set of resolving functions were found according to the structure function. The problem of compensation of distortions caused by turbulence was solved with the aid of a new local-linear super-resolution method. This method allows to resolve turbulent distortions of PSF at low signal-to-noise ratio.

Modelling of propagation and scintillation of a laser beam through atmospheric turbulence

The investigation was fulfilled on the basis of the Navier-Stokes equations for viscous heat-conducting gas. The Helmholtz decomposition of the velocity field into a potential part and a solenoidal one was used. We considered initial vorticity to be small. So the results refer only to weak turbulence. The solution has been represented in the form of power series over the initial vorticity, the coefficients being multiple integrals. In such a manner the system of the Navier- Stokes equations was reduced to a parabolic system with constant coefficients at high derivatives. The first terms of the series are the main ones that determine the properties of acoustic radiation at small vorticity. We modelled turbulence with the aid of an ensemble of vortical structures (vortical rings). Two problems have been considered : (i) density oscillations (and therefore the oscillations of the refractive index) in the case of a single vortex ring; (ii) oscillations in the case of an ensemble of vortex rings (ten in number). We considered vortex rings with helicity, too. The calculations were fulfilled for a wide range of vortex sizes (radii from 0.1 mm to several cm). As shown, density oscillations arise. High-frequency oscillations are modulated by a low-frequency signal. The value of the high frequency remains constant during the whole process excluding its final stage. The amplitude of the low-frequency oscillations grows with time as compared to the high-frequency ones. The low frequency lies within the spectrum of atmospheric turbulent fluctuations, if the radius of the vortex ring is equal to several cm. The value of the high frequency oscillations corresponds satisfactorily to experimental data. The results of the calculations may be used for the modelling of the Gaussian beam propagation through turbulence (including beam distortion, scintillation, beam wandering). A method is set forth which describes the propagation of non-paraxial beams. The method admits generalization to the case of inhomogeneous medium.

Grid turbulence and its interaction with a shock wave

Turbulent fluctuations of density and pressure in air and argon in a shock tube have been investigated as well as their interaction with a shock wave reflected from a perforated plate at the end of a shock tube. Air and argon were used as test gases. The Mach number of the incident shock was 1.9–3.9, that one of the reflected shock was 1.4–2.4. The turbulent length scale behind the incident shock was measured as well as that one behind the reflected shock. The last value is a few times less than the former. It was established that there is overpressure in the turbulent flow behind the reflected shock. The value of the overpressure is 12% in argon and 9% in air.

Density oscillations generated by vortex rings and their effect on scintillation of a Gaussian beam

Density oscillations in the vicinity of vortex rings in air have been investigated. The calculations were fulfilled on the basis of the Navier-Stokes equations. We used series expansions of unknown functions in powers of a parameter which characterizes vorticity. As a result, we got a non-uniform system of parabolic differential equations with constant coefficients. The frequency of oscillations depends only on the dimensions and the shape of the ring in the case of small vorticity (weak turbulence). We analyzed oscillations generated by rings with circular cross-section. The size of the rings varied in a wide range. It includes inertial range and dissipation range. It is interesting to note that first of all the amplitude of oscillations increases, reaches its maximum and then decreases up to zero. These data can be used for modeling the propagation of a Gaussian beam through the turbulent atmosphere. We analyzed intensity fluctuations (scintillations) of the beam after the passage through the non-uniform region which contains vortex rings. We considered an ill-posed problem (that of super-resolution) connected with image restoration. In such cases if the input data are slightly changed, the solution may vary considerably. The proposed procedure is as follows. We change the instrument function in such a manner that it will be reversible one within the limits of accuracy. The procedure enables to solve some problems referring to the turbulent atmosphere. Ke

Pressure fluctuations within a turbulent gas flow and their interaction with a shock wave

The interaction of a shock wave with a turbulent air flow is investigated experimentally. The turbulence was created with the aid of a grid. On its reflection from a perforated disc the wave propagated through a turbulent flow. The Mach number of the incident shock was equal to 1.9–4, the Mach number of the reflected wave was equal to 1.6–2.5. We found the autocorrelation functions of pressure fluctuations and their phase diagrams. The turbulent length scale of pressure fluctuations behind the incident shock was determined. The appropriate quantity behind the reflected wave is less of an order as compared with the previous case. It is established that the pressure behind the reflected wave in the turbulent flow is 7–8% higher as compared with the pressure in the laminar flow, if other conditions are the same.

Statistical properties of density fluctuations in the atmosphere

The improvement of the parametrix method for solving the full system of the Navier-Stokes is presented. As known the fundamental solution equation is an oscillatory one. These oscillations are observed while analyzing the density evolution. Their frequency diminishes as the time grows. The approximate expression is presented for density in the neighborhood of a vortical structure. The laser beam propagation has been analyzed. The method will enable to find time average quantities. We considered the mathematical theory of the laser-schlieren technique. Experimental data on grid-generated turbulence are presented.

Problems related to the beam propagation in the tubulent atmosphere

Related problems are as follows: (i) evolution of the vortical structures which play an important role in turbulence; (ii) laser beam propagation through turbulence; (iii) object-targeting problem. The parametrix method was used. The convergence of the coupled iterative procedure was discussed. We investigated the influence of a point thermal source on the vorticity of a cylindrical vortex. We revised the 3D object-targeting problem.