Turbulent dynamics under two ideal invariants: dynamic phase alignment in plasmas and non-ionized fluids

Turbulent dynamics in the presence of two invariants, e.g., energy and kinetic helicity, is poorly understood in both plasmas and non-ionized fluids. We present results of numerical studies of turbulence in low-$\beta_e$ plasmas at scales below the electron skin depth and turbulence in non-ionized fluids governed by the Navier-Stokes equations. In both systems, the dynamics is dominated by the presence of energy and (generalized) kinetic helicity as two exact invariants. We show that, in the two systems, both invariants are subject to a forward cascade, and we demonstrate that this joint cascade is possible due to the existence of a strong dependence on scale of the Fourier phase alignment angle between, in low-$\beta_e$ plasmas, fluctuations of electric and magnetic potential and, in Navier-Stokes turbulence, fluctuations of velocity and vorticity [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Inertial Alfv\'en Turbulence", Physical Review Letters, 2020]. This phenomenon, termed \textit{dynamic phase alignment}, thus acquires importance as a mechanism regulating the dynamics in the presence of two invariants, arising from their conservation in the joint direct cascade, regardless of the details of the physical interactions [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Navier-Stokes Turbulence", Physical Review Letters, 2021].