Abstract

Hamiltonian Systems and the Generation of Vortex Structures in Fluid Motion

M.V.Fokin (Laboratory of Evolution Equation, Sobolev Institute of Mathematics, Novosibirsk, Russia)

e-mail: fokin@math.nsc.ru

The purpose or this report is to demonstrate new analytical method for studying of the generation of vortex structures in a class of three dimensional incompressible fluid flows. These flows are invariant under the action of a one-parameter symmetry group. It followes in this case that there exists the coordinate system such that the evolution of two of the coordinates is governed by a time-dependent Hamyltonian system with the evolution of remaining coordinate being governed by a first order differential equation that depends only on the other two coordinates and time. The moving vortex structures in the fluid which correspond to local maxima and minima of the Hamyltonian function for given value of time are similar to a tornado. The process of arising, evolution and disappearance of the vortex structures is described. The problem of small oscillations of rotating ideal fluid is considered as the example. The characteristics of spectra in this problem were investigated in details in the case when the velocity and the pressure are dependent only of two spatial variables and the fluid domain is an endless cylinder with the convex domain in the base. The structure of infinite-dimensional manifolds of solutions, which correspond to the various intervals of the energy spectrum, is described. The typical phase portraits of dynamic systems which characterize the oscillations of fluid particles are obtained for each manifold of solutions. The number of vortex structures increases in time and their scale decreases for the motions with continuous energy spectrum. This effect may be considered as one of the mathematical models of development of turbulence. The specific features of this evolution, which correspond to singular continuous spectrum, are characterized.

Section : 7