A zero-mode mechanism for spontaneous symmetry breaking in a turbulent von K\'arm\'an flow

TL;DR

This paper proposes a zero-mode mechanism to explain spontaneous symmetry breaking observed in turbulent flows, linking it to a Goldstone mode and demonstrating it through a toy model with solutions governed by a linear differential equation.

Contribution

It introduces a novel zero-mode mechanism for spontaneous symmetry breaking in turbulence, connecting it to a phase shift and Goldstone mode concepts in a simplified model.

Findings

Spontaneous symmetry breaking occurs at a specific Reynolds number.

Zero-modes obey a Beltrami property.

Fluctuations resemble phonons in turbulence.

Abstract

We suggest that the dynamical spontaneous symmetry breaking reported in a turbulent swirling flow at $Re=40~000$ by Cortet et al., Phys. Rev. Lett., 105, 214501 (2010) can be described through a continuous one parameter family transformation (amounting to a phase shift) of steady states and could be the analogue of the Goldstone mode of the vertical translational symmetry in an ideal system. We investigate a possible mechanism of emergence of such spontaneous symmetry breaking in a toy model of our out-equilibrium system, derived from its equilibrium counterpart. We show that the stationary states are solution of a linear differential equation. For a specific value of the Reynolds number, they are subject to a spontaneous symmetry breaking through a zero-mode mechanism. These zero-modes obey a Beltrami property and their spontaneous fluctuations can be seen as the "phonon of turbulence".