Four-fifths laws in incompressible and magnetized fluids: Helicity, energy and cross-helicity

Abstract

In this paper, we are concerned with the Kolmogorov’s scaling laws of conserved quantities in incompressible fluids. By means of Eyink’s longitudinal structure functions and the analysis of interaction of different physical quantities, we extend celebrated four-fifths laws from energy to helicity in incompressible fluid and, total energy and cross-helicity in magnetohydrodynamic flow. In contrast to pervious 4/5 laws of energy and cross-helicity in magnetized fluids obtained by Politano and Pouquet, they are in terms of the mixed third-order structure functions rather than the structure coupling correlation functions. New insights to the cascade rate of inviscid invariants are provided in incompressible turbulence.