Note on specific chiral ensembles of statistical hydrodynamics: "order function" for transition of turbulence transfer scenarios

TL;DR

This paper investigates how specific chiral ensembles in statistical hydrodynamics exhibit different spectral transfer behaviors based on the shape of the helicity-to-energy ratio, revealing complex dual and multi-directional transfer scenarios.

Contribution

It introduces new insights into how the shape function of helicity and energy spectra influences transfer directions in chiral turbulence ensembles, expanding understanding beyond traditional inverse or forward cascades.

Findings

Different shape functions imply distinct transfer directions.

Simultaneous forward and inverse transfers can occur.

Certain spectral ratios lead to no simple transfer direction.

Abstract

Hydrodynamic helicity signatures the parity symmetry breaking, chirality, of the flow. Statistical hydrodynamics thus respect chirality, as symmetry breaking and restoration are key to their fundamentals, such as the spectral transfer direction and its mechanism. Homochiral sub-system of three-dimensional (3D) Navier-Stokes isotropic turbulence has been numerically realized with helical representation technique to present inverse energy cascade [Biferale et al., Phys. Rev. Lett., {\bf 108}, 164501 (2012)]. The situation is analogous to 2D turbulence where inverse energy cascade, or more generally energy-enstrophy dual cascade scenario, was argued with the help of a negative temperature state of the absolute equilibrium by Kraichnan. Indeed, if the helicity in such a system is taken to be positive without loss of generality, a corresponding negative temperature state can be identified [Zhu et al., J. Fluid Mech., {\bf 739}, 479 (2014)]. Here, for some specific chiral ensembles of turbulence, we show with the corresponding absolute equilibria that even if the helicity distribution over wavenumbers is sign definite, different \textit{ansatzes} of the shape function, defined by the ratio between the specific helicity and energy spectra $s(k)=H(k)/E(k)$, imply distinct transfer directions, and we could have inverse-helicity and forward-energy dual transfers (with, say, $s(k)\propto k^{-2}$ resulting in absolute equilibrium modal spectral density of energy $U(k)=\frac{1}{\alpha +\beta k^{-2}}$, exactly the enstrophy one of two-dimensional Euler by Kraichan), simultaneous forward transfers (with $s(k)=constant$), or even no simply-directed transfer (with, say, non-monotonic $s(k) \propto \sin^2k$), besides the inverse-energy and forward-helicity dual transfers (with, say, $s(k)=k$ as in the homochiral case).