Phase-symmetry breaking as a mechanism for subcritical transition in shell models of turbulence

Subcritical transition to turbulence, in which the laminar state is linearly stable yet finite-amplitude perturbations develop into turbulence, is ubiquitous but predicting it from the structure of the governing equations remains difficult. We demonstrate such a framework using a shell model of turbulence, in which external forcing breaks the phase symmetry of the governing equations. This symmetry breaking suppresses the linear instability of the laminar state, while the energy cascade and spectrum of the developed turbulent state are preserved. A complementary single-triad model admits an exact elliptic neutral stability curve, revealing that the stabilization depends only on the breaking strength and not on the nonlinear coupling coefficients. Since the phase symmetry of the shell model corresponds to Galilean invariance in the Navier-Stokes equations, this mechanism may offer a new perspective on subcritical transition in fluid systems.