Large-scale instabilities of helical flows

Large-scale hydrodynamic instabilities of periodic helical flows are investigated using $3$D Floquet numerical computations. A minimal three-modes analytical model that reproduce and explains some of the full Floquet results is derived. The growth-rate $\sigma$ of the most unstable modes (at small scale, low Reynolds number $Re$ and small wavenumber $q$) is found to scale differently in the presence or absence of anisotropic kinetic alpha (\AKA{}) effect. When an $AKA$ effect is present the scaling $\sigma \propto q\; Re\,$ predicted by the $AKA$ effect theory [U. Frisch, Z. S. She, and P. L. Sulem, Physica D: Nonlinear Phenomena 28, 382 (1987)] is recovered for $Re\ll 1$ as expected (with most of the energy of the unstable mode concentrated in the large scales). However, as $Re$ increases, the growth-rate is found to saturate and most of the energy is found at small scales. In the absence of \AKA{} effect, it is found that flows can still have large-scale instabilities, but with a negative eddy-viscosity scaling $\sigma \propto \nu(b Re^2-1) q^2$. The instability appears only above a critical value of the Reynolds number $Re^{c}$. For values of $Re$ above a second critical value $Re^{c}_{S}$ beyond which small-scale instabilities are present, the growth-rate becomes independent of $q$ and the energy of the perturbation at large scales decreases with scale separation. A simple two-modes model is derived that well describes the behaviors of energy concentration and growth-rates of various unstable flows. In the non-linear regime (at moderate values of $Re$) and in the presence of scale separation, the forcing scale and the largest scales of the system are found to be the most dominant energetically.