Turbulent helicity (velocity–vorticity correlation) represents breaking mirror-symmetry in turbulence. With the aid of an analytical statistical theory for inhomogeneous turbulence, an expression of the Reynolds stress in non-mirror-symmetric turbulence is obtained from the fundamental hydrodynamic equations. It is shown that the helicity gradient enters the Reynolds stress as the coupling coefficient of the mean absolute vorticity Ω* (system rotation and mean relative vorticity). Using this analytical expression, a turbulence model (helicity model) is constructed, which is validated by direct numerical simulations (DNS) of rotating turbulence with helical forcing. This result implies that inhomogeneous turbulent helicity coupled with the mean absolute vorticity Ω* may induce a global flow in the direction of Ω*. This effect is generic in rotating turbulence with inhomogeneous helicity, and is expected to be relevant to several astro- and geo-physical flows, which include cyclones and solar convective motion. The angular-momentum transport in the solar convection zone is discussed from the view point of this helicity effect.

Helicities as well as the turbulent energies are key players of the dynamo process. From viewpoint of turbulent transport, helicities mainly suppress the effective transports. In the presence of inhomogeneous large-scale flow, the turbulent cross helicity (cross-correlation between the velocity and magnetic-field fluctuations) enters the turbulent electromotive force and its suppression/generation effect tends to be balanced with the turbulent magnetic diffusivity. At the same time, the turbulent cross helicity coupled with the mean magnetic strain affects the momentum transport. In addition to these effects of helicities, in strong compressible magnetohydrodynamic (MHD) turbulence, the density variance contributes to the turbulent electromotive force as the coupling coefficient of the obliqueness of the mean magnetic field to the density gradient. This means that the density variance as well as the helicities alters the turbulent transport. This density-variance effect is expected to enhance the intensity of turbulence across the slow MHD shock, which may contribute to the realization of a localized fast magnetic reconnection.