Cascade time-scales for energy and helicity in homogeneous isotropic turbulence

Abstract

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 $τ_H$ for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this timescale using the phenomenological arguments used by Kraichnan to derive the timescale $τ_E$ for energy transfer (J. Fluid Mech. {\bf 47}, 525--535 (1971)). We show that in general $τ_H$ may not be neglected compared to $τ_E$, 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 $τ_E$ in the low wavenumbers with both energy and helicity spectra scaling as $k^{-5/3}$; and by $τ_H$ at larger wavenumbers 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 wavenumber $k_h$ which is less than the Kolmogorov dissipation wavenumber, at which both energy and helicity cascades terminate due to dissipation effects. Data from direct numerical simulations are used to check our predictions.