The effect of helicity on the correlation time of large scale turbulent flows

Solutions of the forced Navier-Stokes equation have been conjectured to thermalize at scales larger than the forcing scale, similar to an absolute equilibrium obtained for the spectrally-truncated Euler equation. Using direct numeric simulations of Taylor-Green flows and general-periodic helical flows, we present results on the probability density function, energy spectrum, auto-correlation function and correlation time that compare the two systems. In the case of highly helical flows, we derive an analytic expression describing the correlation time for the absolute equilibrium of helical flows that is different from the $E^{-1/2}k^{-1}$-scaling law of weakly helical flows. This model predicts a new helicity-based scaling law for the correlation time as $\tau(k)\sim H^{-1/2}k^{-1/2}$. This scaling law is verified in simulations of the truncated Euler equation. In simulations of the Navier-Stokes equations the large scale modes of forced Taylor-Green symmetric flows (with zero total helicity and large separation of scales) follow the same properties as absolute equilibrium including a $\tau(k) \sim E^{-1/2}k^{-1}$ scaling for the correlation time. General-periodic helical flows also show similarities between the two systems, however the largest scales of the forced flows deviate from the absolute equilibrium solutions.