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

Abstract

Subcritical transition to turbulence, in which the laminar state is linearly stable yet finite-amplitude perturbations develop into turbulence, is ubiquitous but lacks a simple analytical framework. 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.

Figures (3)

• Figure 1: (a) Neutral stability curve [Eq. \ref{['eq:oval']}] for the ST model in the $(\nu,\, |U_+|)$ plane. LU: linearly unstable; GS: globally stable (bounded by the energy stability threshold $\nu_e$); LSGU: linearly stable but globally unstable. (b) Eigenvalues of the linear stability problem for the shell model at $U = 0$ (dashed) and $U = 5$ (crosses), compared with the asymptotic form Eq. \ref{['eq:asym']} (circles).
• Figure 2: Disturbance energy $E_d(t)$ for various initial amplitudes $h$. (a) ST model with $\nu = 10^{-3}$: solid lines, $U = 0.6 > U_c$; dashed lines, reference system ($U = 0$). (b) Shell model with $\nu = 10^{-6}$: solid lines, $U = 0.8 > U_c$; dashed lines, reference system ($U = 0$). In both cases, small disturbances decay in the gauge-equivalent system while large disturbances sustain nonlinear states.
• Figure 3: (a) Energy flux $-\Pi(n)/\epsilon$ and dissipation $D(n)/\epsilon$ for the shell model. (b) Energy spectrum $\epsilon^{-2/3}k_d^{2/3}E$. Plus markers: gauge-equivalent system (GE), $(U_{N_f+3m},\,U_{N_f+1+3m},\,U_{N_f+2+3m}) = (U,\,U,\,-2U)$; crosses: gauge-inequivalent system (GI), $(U_{N_f+3m},\,U_{N_f+1+3m},\,U_{N_f+2+3m}) = (U,\,U,\,0)$. Dashed lines: reference system ($U=0$). Vertical dotted lines mark $k/k_d = 10^{-3}$ and $1$.