  Generalized Navier-Stokes (GNS) equations describing three-dimensional (3D)
active fluids with flow-dependent spectral forcing have been shown to possess
numerical solutions that can sustain significant energy transfer to larger
scales by realising chiral Beltrami-type chaotic flows. To rationalise these
findings, we study here the triad truncations of polynomial and Gaussian GNS
models focusing on modes lying in the energy injection range. Identifying a
previously unknown cubic invariant, we show that the asymptotic triad dynamics
reduces to that of a forced rigid body coupled to a particle moving in a
magnetic field. This analogy allows us to classify triadic interactions by
their asymptotic stability: unstable triads correspond to rigid-body forcing
along the largest and smallest principal axes, whereas stable triads arise from
forcing along the middle axis. Analysis of the polynomial GNS model reveals
that unstable triads induce exponential growth of energy and helicity, whereas
stable triads develop a limit cycle of bounded energy and helicity. This
suggests that the unstable triads dominate the initial relaxation stage of the
full hydrodynamic equations, whereas the stable triads determine the
statistically stationary state. To test this hypothesis, we introduce and
investigate the Gaussian active turbulence model, which develops a
Kolmogorov-type $-5/3$ energy spectrum at large wavelengths. Similar to the
polynomial case, the steady-state chaotic flows spontaneously accumulate
non-zero mean helicity while exhibiting Beltrami statistics and upward energy
transport. Our results suggest that self-sustained Beltrami-type flows and an
inverse energy cascade may be generic features of 3D active turbulence models
with flow-dependent spectral forcing.
