Theory for helical turbulence under fast rotation

Recent numerical simulations have shown the strong impact of helicity on homogeneous rotating hydrodynamic turbulence. The main effect can be summarized through the law $n+\tilde{n}=-4$, where $n$ and $\tilde{n}$ are the power law indices of the one-dimensional energy and helicity spectra, respectively. We investigate this rotating turbulence problem in the small Rossby number limit by using the asymptotic weak turbulence theory derived previously. We show that the empirical law is an exact solution of the helicity equation where the power law indices correspond to perpendicular (to the rotation axis) wave number spectra. It is proposed that when the cascade towards small scales tends to be dominated by the helicity flux the solution tends to $\tilde{n}=-2$, whereas it is $\tilde{n}=-3/2$ when the energy flux dominates. The latter is compatible with the solution previously observed numerically and derived theoretically in the weak turbulence regime when only the energy equation is used, whereas the former solution is constrained by a locality condition.