Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation : The case of the sudden expansion
Some nonlinear dynamical systems are metastable : under weak noise, they randomly switch between two configurations after long, unpredictable times. This occurs, for instance, in quasistationary turbulent zonal jets. However, because the mean transition time between metastable states is extremely long, such behavior is rarely observed through standard high-fidelity numerical simulations. Instead, rare event algorithms based on large-deviation theory are often used. To compute flow statistics more efficiently, we develop a multiple-scale weakly nonlinear expansion of the Navier-Stokes equations. These are forced by weak, narrow-band noise acting on the same slow timescale as the amplitude of the dominant mode near a bifurcation. For steady symmetry-breaking bifurcations, we rigorously derive a stochastically forced Stuart-Landau equation for the slow amplitude of the symmetry-breaking mode. All coefficients are computed from scalar products of linearly obtained fields. We test the validity of this reduced-order model on flow past a sudden expansion, post-bifurcation. Because the amplitude equation is potential-driven, the solution’s probability density function is easily obtained via the Fokker-Planck equation. This method yields accurate flow statistics at a fraction of the numerical cost, matching those from long-time direct simulations of the stochastically forced Navier-Stokes equations for various noise levels. I will also discuss the derivation of an amplitude equation which extends the linear stochastic response of linearly stable but strongly nonnormal systems like shear layers or jets in the weakly nonlinear realm.
Contact Nathanaël Machicoane for more information or to schedule a discussion with the seminar speaker.