Phase dynamics and their role determining energy flux in hydrodynamic shell models

TL;DR

This paper investigates how phase dynamics influence energy flux in hydrodynamic shell models, revealing that phase interactions determine cascade directions and energy transfer, with analytical predictions validated by simulations.

Contribution

It introduces a simplified phase oscillator model to analytically predict energy flux and demonstrates how phase dynamics govern cascade directions in shell models.

Findings

All energy-conserving shell models with sign-indefinite quadratic quantities have a forward cascade.

Phase dynamics prevent inverse energy cascades in certain shell models.

Analytical phase statistics match numerical shell model simulations.

Abstract

The transfer of energy and other conserved quantities across scales, also known as flux or spectral flux, is a central aspect of out-of-equilibrium systems such as turbulent hydrodynamic flows. Despite its role in the few predictive theories that exist, a dynamical understanding of what determines said flux (and its direction in scale) has yet to be established. In this study, we work towards this understanding by investigating how the dynamics of complex Fourier velocity phases influence the flux of conserved quantities in hydrodynamic shell models. The phase dynamics, like energy evolution, are influenced by contributions from all neighboring triads, making the full problem intractable. Instead, we assume that the dynamics of the triad phases are determined solely by the so-called self-interaction term and treat the other neighboring triad terms as noise. This transforms the phase dynamics into that of a noisy phase oscillator, which we solve analytically to predict phase statistics. We validate this assumption with a suite of shell model simulations. Our results give us analytical predictions for the energy flux, when the energy spectrum is given. We prove that all shell models that conserve energy along with a sign-indefinite quadratic quantity (this includes three-dimensional turbulence analogues) undergo a forward energy cascade, and further show that the phase dynamics prevent the shell model analogue of two-dimensional turbulence from forming an inverse cascade of energy.