Conservation Laws of Helically Symmetric Flows and Their Importance for Turbulence Theory

Abstract

Our present understanding of statistical 3D turbulence dynamics in the large wave number limit (or small scales) largely relies on the dissipation of turbulent kinetic energy a quantity which is invariant under all symmetry groups of Navier-Stokes equations except the scaling groups. In turn, this implies Kolmogorov’s sub-range theory and to a large part our understanding of energy transfer. On the other hand in 2D turbulence, which is translational invariant in one direction, the transfer mechanism among scales is rather different since the vortex stretching mechanism is non-existing. Instead, the scale determining key invariant is enstrophy: an area integral of the vorticity squared which is one of the infinite many integral invariants (Casimirs) of 2D inviscid fluid mechanics. Hence the basic transfer mechanisms between 2D and 3D turbulence are very different. To close this gap we consider flows with a helical symmetry which is a twist of translational and rotational symmetry. The resulting equations are “2$\frac{1}{2}$ D” which means they have three independent velocity components though only two independent spatial variables. We presently show that in the inviscid limit the helically symmetric equations of motion admit a finite number of new non-trivial conservation laws comprising

∙ vorticity - though the basic vortex stretching mechanism is still active for helical flows and

∙ stream function even in a non-linear form clearly stating a non-local conservation laws since the stream-function is a line integral.

It is to be expected that the new conservation laws may give some deeper insight into turbulence dynamics and hence bridging 2D and 3D turbulence.