The circulation density and its role in 3D turbulence

✍ Scribed by Sergey V. Nazarenko

• Book ID: 104297081
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 354 KB
• Volume: 102
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/s0167-2789(96)00180-7

✦ Synopsis

The scaling argument applied to the vorticity cascade in 2D turbulence results in the well-known k -3 energy spectrum. Kelvin theorem for the velocity circulation generalizes the vorticity conservation law on the 3D fluids. Using the Kelvin theorem and the incompressibility of the fluid one can derive Lagrangian conservation of a quantity having the physical meaning of a circulation density. Integrated over the entire space, the powers of the circulation density form a series of the motion integrals which coincides with the vorticity series in the 2D limit. We will discuss the question of applicability of the scaling arguments to the cascades of the geometrical integrals in 3D turbulence. We will see that the cascades of all but one circulation integrals are ruled out by the reconnection kinematics. The only exception is the integral corresponding to the total volume of the vortex tubes, whose cascade corresponds to the k -3 energy spectrum. Relation of the circulation density to the Clebsch variables will be considered. We will see that the circulation density can be chosen as one of the Clebsch variables. In the case of the stationary flows, such Clebsch variables becomes an action-angle pair.