Transport and Wave Propagation in Turbulence

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This project concerns the transport of scalars, vectors, and waves in developed turbulence. The first part of the project deals with the statistical modeling of turbulent velocity and methods of analyzing the statistics of scalar (for example, temperature) fluctuations. A characteristic feature of turbulent velocity is the presence of a wide range of scales. This induces a similarly wide range of scales in fluctuations of temperature and hence in the local fluctuations of, for example, the refractive index. The study of the refractive index in the turbulent air is an important challenging problem for mathematical analysis as well as technological applications. The second part of the project addresses the statistical properties of wave motion in atmospheric turbulence. In particular, the project focuses on several regimes where simplified description of waves can be derived, some of which are known to be widely applicable. They include the parabolic approximation, the white-noise approximation, and the radiative transfer limit. The goal is to formulate a mathematical framework to study the various approximations in a rigorous and unified way. The details of the solutions in the various approximations will be investigated, providing guides for the design of communications and imaging systems.Understanding of scalar and wave transport through atmospheric turbulence is essential for many important applications, including heat and contaminant transport, laser and wireless communications, remote sensing, and imaging. In the case of sound or optical wave propagation in the atmosphere, for instance, the wave speed undergoes random fluctuations determined by small-scale temperature and humidity fluctuations. This project develops a unified mathematical framework to address problems of scalar transport and wave propagation through turbulent media. The results of the research will contribute to the improvement of optical communications and imaging technologies.