The reflection-antisymmetric counterpart of the Kármán–Howarth dynamical equation

We study the isotropic, helical component in homogeneous turbulence using statistical objects which have the correct symmetry and parity properties. Using these objects we derive an analogue of the Kármán–Howarth equation, that arises due to lack of mirror-reflection-symmetry in isotropic flows. The main equation we obtain is consistent with the results of Chkhetiani [JETP 63 (1996) 768] and L’vov et al. [Exact result for the 3rd order correlations of velocity in turbulence with helicity, 1997. http://xxx.lanl.gov/abs/chao-dyn/9705016] but is derived using only velocity correlations, with no direct consideration of the vorticity or helicity. This alternative formulation offers an advantage to both experimental and numerical measurements. We also postulate, under the assumption of self-similarity, the existence of a hierarchy of scaling exponents for helical velocity correlation functions of arbitrary order, analogous to the Kolmogorov prediction for the scaling exponents of velocity structure function.