Passive scalar cascades in rotating helical and non-helical flows

We study how helicity affects the spectrum of a passive scalar in rotating turbulent flows, using numerical simulations of turbulent flows with or without rotation, and with or without injection of helicity. Scaling laws for energy and passive scalar spectra in the direction perpendicular to the rotation axis differ in rotating helical flows from the ones found in the non-helical case, with the spectrum of passive scalar variance in the former case being shallower than in the latter. A simple phenomenological model that links the effects of helicity on the energy spectrum with the passive scalar spectrum is presented.

💡 Research Summary

This paper investigates how helicity influences the cascade of a passive scalar in rotating turbulent flows by means of high‑resolution direct numerical simulations (DNS). Four distinct configurations are examined: non‑rotating non‑helical, non‑rotating helical, rotating non‑helical, and rotating helical. The governing equations are the incompressible Navier–Stokes equations coupled with an advection‑diffusion equation for a passive scalar. Rotation is introduced through a uniform Coriolis term Ω × u, while helicity is injected by a forcing that correlates velocity and vorticity, thereby controlling the helicity injection rate σ. All runs are performed on a 1024³ periodic grid, with Reynolds numbers of order 10⁴ and Rossby numbers ranging from O(1) (weak rotation) down to O(0.1) (strong rotation). Energy and scalar forcing are applied at large scales (k≈2–3) to achieve statistically stationary states.

The main findings can be summarized as follows. In the absence of rotation, helicity does not alter the classic Kolmogorov k⁻⁵ᐟ³ energy spectrum nor the Obukhov–Corrsin k⁻⁵ᐟ³ passive‑scalar spectrum; both helical and non‑helical runs collapse onto the same scaling. When rotation is present, the flow becomes strongly anisotropic, with dynamics dominated by motions perpendicular to the rotation axis (k⊥). For rotating non‑helical turbulence the kinetic energy spectrum steepens to approximately k⊥⁻², while the scalar spectrum follows a steeper k⊥⁻⁵ᐟ³ law, reflecting the suppression of forward transfer by the Coriolis force.

In contrast, rotating helical turbulence exhibits a markedly different behavior. The kinetic energy spectrum becomes even steeper (k⊥⁻²·² to k⊥⁻²·⁵), indicating that helicity enhances the forward cascade of kinetic energy despite the rotational constraint. Simultaneously, the passive‑scalar variance spectrum becomes shallower, with an exponent close to –3/2 rather than –5/3. This implies that helicity prolongs the scalar’s residence time at larger scales, allowing more variance to accumulate at intermediate wavenumbers.

To rationalize these observations, the authors propose a phenomenological model that modifies the characteristic transfer time τ(k) by incorporating helicity. In non‑helical rotating turbulence τ_R(k)∼Ω⁻¹k⁻¹ᐟ², reflecting the inertial‑wave mediated transfer. Helicity introduces an additional factor proportional to 1+α H(k)/E(k), where H(k) is the helicity spectrum, E(k) the kinetic energy spectrum, and α a dimensionless constant calibrated from the DNS data. The modified transfer time τ(k)=τ_R(k)