Sign-Indefinite Helicity and the Structure of Weak Turbulence in Inertial and Non-Hermitian Waves
We investigate how sign-indefinite quadratic invariants shape turbulent cascades in incompressible flows with broken time-reversal symmetry, where the dynamics supports strongly anisotropic dispersive waves. Focusing on rotating Euler flow and odd-viscous Euler flow, we isolate the wave component study the corresponding weak-turbulence kinetic equation. We show that helicity conservation substantially simplifies the kinetic equation. Fixing the energy flux by a natural gauge choice, we identify the turbulent spectrum as the unique scale-invariant solution that sustains a constant flux of energy from large to small scales. Under a mild approximation motivated by the accumulation of energy near slow modes, we compute the leading angular dependence and uncover an integrable singularity along the slow-mode curve, that agrees with previous results. We then demonstrate that helicity reorganizes cascade directions at the level of resonant triads. Although helicity is globally sign-indefinite, the helical decomposition splits it into sign-definite contributions on each polarization branch. Triads whose three legs lie on the same branch behave as if constrained by a sign-definite invariant and drive an upscale transfer of energy, producing systematic backscatter even when the net cascade is direct. In the helicity-definite limit (single-branch dynamics), the kinetic equation admits an additional scale-invariant solution associated with helicity transport. Finally, we validate the analytical predictions by numerically evaluating the collision integral in the strongly anisotropic limit, revealing a family of stationary solutions in that regime.