Decay of helicity in homogeneous turbulence

The self-similar relaxation of helicity in homogeneous turbulence has been considered taking into account integral invariants $\int \limits_0^\infty r^{m} \langle \mathbf{u}(\mathbf{x})\boldsymbol{\omega } (\mathbf{x}+\mathbf{r})\rangle dr=I_{m}^{h}$ (where $\boldsymbol{\omega }=\text{rot}\mathbf{u}$ and $r=|\mathbf{r}|$). It has been shown that integral invariants with $m = 3$ for both helicity and energy are possible in addition to helical analogs of Loitsyanskii ($m = 4$) and Birkhoff-Saffman ($m = 2$) invariants associated with the conservation laws of momentum and angular momentum, respectively. Helicity always relaxes more rapidly than the energy. Its decay exponent is in the interval from $-3/2$ to $-5/2$ versus the interval from $-6/5$ to $-10/7$ for the energy.