THE CIRCULATION DENSITY AND ITS ROLE IN 3D TURBULENCE

The scaling argument applied to the vorticity cascade in 2D turbulence results in the well-known k(-3) energy spectrum. Kelvin theorem for t he velocity circulation generalizes the vorticity conservation law on the 3D fluids. Using the Kelvin theorem and the incompressibility of t he fluid one can derive Lagrangian conservation of a quantity having t he physical meaning of a circulation density. Integrated over the enti re space, the powers of the circulation density form a series of the m otion integrals which coincides with the vorticity series in the 2D li mit. We will discuss the question of applicability of the scaling argu ments to the cascades of the geometrical integrals in 3D turbulence. W e will see that the cascades of all but one circulation integrals are ruled out by the reconnection kinematics. The only exception is the in tegral corresponding to the total volume of the vortex tubes, whose ca scade corresponds to the k(-3) energy spectrum. Relation of the circul ation density to the Clebsch variables will be considered. We will see that the circulation density can be chosen as one of the Clebsch vari ables. In the case of the stationary flows, such Clebsch variables bec omes an action-angle pair.