Quasi-two Dimensional Hydrodynamics and Interaction of Vortex Tubes

This paper is long overdue. The most of the results presented here were obtained in 1986-87. Just a small portion of them (the equations for dynamics of the pair of counter-rotating vortices and their self-similar solutions) were published in time in 1988 [1]. The publication was very brief and did not include any details of the calculations. Nevertheless, it was noticed and then generously cited by R. Klein, A. Maida and K. Damodaran [2]. In this publication I would like to express my gratitude to these authors who reobtained an essential part of the results published below. In spite of the fact that some of the results published below (equations for the systems of almost parallel vortices) could be found in their article, I believe that my paper deserves to be published. Some of the results presented here are completely new, and, which is more important, the methodology published here is completely different from one used in [2]. In this article we develop a systematic approach to stydu stationary and nonstationary flows of ideal incompressivle fluid under assumption that the gradients in one preferred direction z are much less than the gradients in the orthogonal plane. Such flows could be called quazi-two dimentional. There are two motivations for paying a special attention to this class of fluid motion. One is connected with the classical problem of the ”blow-up” in the Euler equation. According to the most plausible scenario, (see, for instance, [3]), in the point of blow-up the vorticity becomes infinite. As far as vorticity is a vector, this assumption presumes that the flow near the blow-up point is almost two-dimentional and the velocity field is concentrated mostly in the plane orthogonal to the vorticity vector. An elaboration of this regular tool for description of this type of flow looks very timely. Another motivation is the vortex dynamics. This is a subject which has a chance to become the backbone of the future theory of turbulence. Probably, there is no way to explain qualitatively and quantitatively the fundamental phenomenon of intermittency but a careful study of the dynamics of the vortex tubes or their systems in a real 3-dimentional nonstationary flow. ”Vortices are the sinews of turbulence” said K. Moffatt (look at his lecture on the Seventh European Turbulence Conference [4]).