Statistical mechanics of vortex lines

Statistical mechanics of three-dimensional flows of an ideal incompressible fluid is considered. An ideal fluid differs from the usual Hamiltonian systems of statistical mechanics by possessing an infinite number of integrals of motion that are circulations of velocity over closed fluid contours. To reduce the problem to a standard one, the governing equations should be written in a Hamiltonian form in which all integrals of motion other than energy are eliminated. This is achieved by a generalization of the variational principle, which solves this problem in a two-dimensional case. The formulated variational principle can be interpreted as a variational principle for the dynamics of vortex lines. An invariant measure in the space of vortex lines is derived. For effectively two-dimensional flows this measure is reduced to the invariant measure obtained previously. As an example of application to effectively three-dimensional flows, the equation for averaged stream function for turbulent flow in pipes is derived.