Verify parallel-direction scaling in odd wave turbulence

Determine whether the anisotropic kinetic energy spectrum E(k_perp, k_parallel) of weak odd-wave turbulence in the three-dimensional incompressible Navier–Stokes fluid with odd viscosity exhibits the predicted parallel-direction scaling E ∝ k_parallel^{-1/2} (as part of the Kolmogorov–Zakharov spectrum E ∝ k_perp^{-3/2} k_parallel^{-1/2}) by establishing this scaling over a sufficiently broad inertial range in k_parallel, where the parallel energy flux is small.

Background

The paper derives a kinetic equation for odd wave turbulence in a three-dimensional incompressible fluid with odd viscosity and obtains an exact anisotropic Kolmogorov–Zakharov spectrum E(k_perp, k_parallel) ∝ k_perp^{-3/2} k_parallel^{-1/2}. Direct numerical simulations are used to validate weak wave turbulence signatures and the predicted perpendicular scaling.

While the simulations confirm the perpendicular-direction predictions and the presence of a direct cascade, the authors report that the inertial range in the parallel direction is too narrow due to a very small energy flux, preventing a numerical test of the predicted k_parallel^{-1/2} scaling. Establishing this parallel scaling remains an explicit unresolved task within the paper.