Dynamic Phase Alignment in Navier-Stokes Turbulence*

In Navier-Stokes turbulence, energy and helicity injected at large scales are subject to a joint direct cascade, with both quantities exhibiting a spectral scaling $\sim k^{-5/3}$. A ``na\"ive", dimensional estimate of the spectrum of helicity would, however, yield the scaling $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \sim k^{-2/3}$, violating conservation of the helicity flux in the inertial range. We demonstrate via direct numerical simulations that this apparent contradiction is revolved because of the existence of a strong scale-dependent Fourier phase alignment between velocity and vorticity fluctuations, with the phase alignment angle scaling as $\cos\alpha_k\propto k^{-1}$ [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Navier-Stokes Turbulence", Physical Review Letters, 2021]. This strong dependence on scale of $\cos\alpha_k$, termed \textit{dynamic phase alignment}, underpins the spectral scaling of helicity, i.e., $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \cos\alpha_k \sim k^{-5/3}$. Dynamic phase alignment plays a role in the turbulent dynamics in the presence of two invariants beyond Navier-Stokes, and it has been shown to underpin the joint direct cascade of energy and (generalized) helicity in a variety of turbulent plasma environments [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Inertial Alfv\'en Turbulence", Physical Review Letters, 2020].