Helicity — the integral of velocity dotted with vorticity — measures the knottedness of a flow. In three-dimensional turbulence, it is approximately conserved alongside energy, and its sign determines whether energy cascades forward or inverse. But helicity is globally sign-indefinite: positive in some regions, negative in others, with no guarantee about the total.
Aharony Shapira and Shavit (arXiv:2603.14181) show that this global sign-indefiniteness conceals a cleaner structure. In rotating flows and flows with odd viscosity, the linearized waves decompose into polarization branches — inertial waves of different handedness. On each branch, helicity is sign-definite. The globally tangled quantity becomes locally clean when projected onto the right basis.
This decomposition simplifies the weak turbulence kinetic equation. The collision integral — which describes how wave modes exchange energy through three-wave interactions — becomes tractable when each mode carries definite helicity sign. The cascade direction at each triad is determined by the helicity signs of the participating modes, not by the global helicity budget.
The structural insight is that sign-indefiniteness is a property of the wrong decomposition, not of the physics. The total helicity integrates contributions from branches with opposite signs. The cancellation obscures what each branch is doing individually. Projecting onto polarization modes resolves the ambiguity — not by introducing new physics, but by choosing coordinates that respect the wave structure.
For odd-viscous flows — fluids where the viscosity tensor has antisymmetric components, as in magnetized plasmas and rotating superfluids — the non-Hermitian wave operator produces complex eigenvalues that break time-reversal symmetry at the linear level. The helicity decomposition reveals how this broken symmetry selects preferred cascade directions in each polarization branch independently.