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

We derive a time-scale $\tau_H$ for energy and helicity transfer in homogeneous, isotropic turbulence from the helical co-spectrum, using the arguments of Kraichnan (J. Fluid Mech. {\bf 47}, 525--535 (1971). While the Kraichnan time-scale, $\tau_E$ depends on in-plane shearing motions derived from reflection-symmetric velocity correlations, $\tau_H$ depends on out-of-plane shearing motions introduced by the presence of helicity. We show that in general $\tau_H$ may not be neglected even for rather low relative helicity. We postulate an inertial range joint cascade in which the dynamics are dominated by $\tau_E$ in the low wavenumbers with spectra scaling as $k^{-5/3}$; and by $\tau_H$ at larger wavenumbers with spectra scaling as $k^{-4/3}$. The commonly observed 'bottleneck' in the energy spectrum might be explained by such a two-stage cascade. We derive a wavenumber $k_h$ which is less than the Kolmogorov dissipation wavenumber, at which both energy and helicity cascades terminate dur to dissipation effects. Data from direct numerical simulations are used to check our predictions.