Cascade time scales for energy and helicity in homogeneous isotropic turbulence

We extend the Kolmogorov phenomenology for the scaling of energy spectra in high-Reynolds-number turbulence, to explicitly include the effect of helicity. There exists a time scale ${\tau }_H$ for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this time scale using the phenomenological arguments used by Kraichnan to derive the time scale ${\tau }_E$ for energy transfer [R. H. Kraichnan, J. Fluid Mech. 47, 525 (1971)]. We show that in general ${\tau }_H$ may not be neglected compared to ${\tau }_E$, even for rather low relative helicity. We then deduce an inertial range joint cascade of energy and helicity in which the dynamics are dominated by ${\tau }_E$ in the low wave numbers with both energy and helicity spectra scaling as $k^{-5∕3}$; and by ${\tau }_H$ at larger wave numbers with spectra scaling as $k^{-4∕3}$. We demonstrate how, within this phenomenology, the commonly observed “bottleneck” in the energy spectrum might be explained. We derive a wave number $k_h$ which is less than the Kolmogorov dissipation wave number, at which both energy and helicity cascades terminate due to dissipation effects. Data from direct numerical simulations are used to check our predictions.