Sign-Indefinite Helicity and the Structure of Weak Turbulence in Inertial and Non-Hermitian Waves

Abstract

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.

Figures (3)

• Figure 1: Logarithm of the energy spectrum $e_k=C_0k^{-3}|\cos\theta_k|^{-1/2}$ on the $(\rho:=\sqrt{k_x^2+k_y^2},k_z)$ plane, outside an angular source centered at radius $k_f$ and emitting a radial energy flux $\Pi^r(\theta_k)=\Pi_0 |\cos\theta_k|^{-1} \hat{k}/ k$.
• Figure 2: Set of zeros of the collision integral, found numerically Roots.jl in terms of the angular $f_0$ and radial $w_0$ power-laws. We highlight the energy cascade solution: $w_0=3,\ f_0=0.5$.
• Figure 3: Radial Fluxes of energy and helicity, as given by Eq. \ref{['eq:fluxang']} (See appendix for the explicit form), as a function of $k_\alpha^z$, evaluated at $k_\alpha=1$. The fluxes were numerically integrated HCubature and normalized to the total energy flux (blue line). Error regions correspond to $\pm 100$ error estimation of the numerical integration.